The Weibull Distribution

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++.

Weibull Probability Density Function

The 3-Parameter Weibull Distribution

The 3-parameter Weibull pdf is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t-\gamma }{\eta }}\right) ^{\beta }} \,\! }[/math]

where:

[math]\displaystyle{ f(t)\geq 0,\text{ }t\geq \gamma \,\! }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math]

[math]\displaystyle{ \eta \gt 0 \,\! }[/math]

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and:

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope)

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life)

The 2-Parameter Weibull Distribution

The 2-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\! }[/math]

The 1-Parameter Weibull Distribution

The 1-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.

Weibull Statistical Properties

The Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} \,\! }[/math], (also called MTTF of the Weibull pdf is given by:

[math]\displaystyle{ \overline{T}=\gamma +\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

where

[math]\displaystyle{ \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

is the gamma function evaluated at the value of

[math]\displaystyle{ \left( { \frac{1}{\beta }}+1\right) \,\! }[/math].

The gamma function is defined as:

[math]\displaystyle{ \Gamma (n)=\int_{0}^{\infty }e^{-x}x^{n-1}dx \,\! }[/math]

For the 2-parameter case, this can be reduced to:

[math]\displaystyle{ \overline{T}=\eta \cdot \Gamma \left( {\frac{1}{\beta }}+1\right) \,\! }[/math]

Note that some practitioners erroneously assume that [math]\displaystyle{ \eta \,\! }[/math] is equal to the MTTF, [math]\displaystyle{ \overline{T}\,\! }[/math]. This is only true for the case of [math]\displaystyle{ \beta=1 \,\! }[/math] or

[math]\displaystyle{ \begin{align} \overline{T} &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {\frac{1}{1}}+1\right) \\ &= \eta \cdot \Gamma \left( {2}\right) \\ &= \eta \cdot 1\\ &= \eta \end{align} }[/math]

The Median

The median, [math]\displaystyle{ \breve{T} }[/math], of the Weibull distribution is given by:

[math]\displaystyle{ \breve{T}=\gamma +\eta \left( \ln 2\right) ^{\frac{1}{\beta }} }[/math]

The Mode

The mode, [math]\displaystyle{ \tilde{T} }[/math], is given by:

[math]\displaystyle{ \tilde{T}=\gamma +\eta \left( 1-\frac{1}{\beta }\right) ^{\frac{1}{\beta }} }[/math]

The Standard Deviation

The standard deviation, σ_T, is given by:

[math]\displaystyle{ \sigma _{T}=\eta \cdot \sqrt{\Gamma \left( {\frac{2}{\beta }}+1\right) -\Gamma \left( {\frac{1}{ \beta }}+1\right) ^{2}} }[/math]

The Weibull Reliability Function

The equation for the 3-parameter Weibull cumulative density function, cdf, is given by:

[math]\displaystyle{ F(t)=1-e^{-\left( \frac{t-\gamma }{\eta }\right) ^{\beta }} }[/math].

This is also referred to as unreliability and designated as [math]\displaystyle{ Q(t) \,\! }[/math] by some authors.

Recalling that the reliability function of a distribution is simply one minus the cdf, the reliability function for the 3-parameter Weibull distribution is then given by:

[math]\displaystyle{ R(t)=e^{-\left( { \frac{t-\gamma }{\eta }}\right) ^{\beta }} }[/math]

The Weibull Conditional Reliability Function

The 3-parameter Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)={ \frac{R(T+t)}{R(T)}}={\frac{e^{-\left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }}}{e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }}}} }[/math]

or:

[math]\displaystyle{ R(t|T)=e^{-\left[ \left( {\frac{T+t-\gamma }{\eta }}\right) ^{\beta }-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }\right] } }[/math]

These gives the reliability for a new mission of [math]\displaystyle{ t \,\! }[/math] duration, having already accumulated [math]\displaystyle{ T \,\! }[/math] time of operation up to the start of this new mission, and the units are checked out to assure that they will start the next mission successfully. It is called conditional because you can calculate the reliability of a new mission based on the fact that the unit or units already accumulated hours of operation successfully.

The Weibull Reliable Life

The reliable life, [math]\displaystyle{ T_{R} \,\! }[/math], of a unit for a specified reliability, R, starting the mission at age zero, is given by:

[math]\displaystyle{ T_{R}=\gamma +\eta \cdot \left\{ -\ln ( R ) \right\} ^{ \frac{1}{\beta }} }[/math]

This is the life for which the unit/item will be functioning successfully with a reliability of R. If R=0.50, then [math]\displaystyle{ T_{R}=\breve{T} }[/math], the median life, or the life by which half of the units will survive.

The Weibull Failure Rate Function

The Weibull failure rate function, [math]\displaystyle{ \lambda(t) \,\! }[/math], is given by:

[math]\displaystyle{ \lambda \left( t\right) = \frac{f\left( t\right) }{R\left( t\right) }=\frac{\beta }{\eta }\left( \frac{ t-\gamma }{\eta }\right) ^{\beta -1} }[/math]

Characteristics of the Weibull Distribution

As was mentioned previously, the Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, β, and the scale parameter, η, affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, i.e. the 3-parameter form. The appropriate substitutions to obtain the other forms, such as the 2-parameter form where γ = 0, or the 1-parameter form where β = C = constant, can easily be made.

Characteristic Effects of the Shape Parameter, β

The Weibull shape parameter, β, is also known as the slope. This is because the value of β is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when β = 1, the of the 3-parameter Weibull reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = }[/math] failure rate. The parameter β is a pure number, i.e. it is dimensionless.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

The Weibull distribution is widely used in reliability and life data analysis due to its versatility. Depending on the values of the parameters, the Weibull distribution can be used to model a variety of life behaviors. We will now examine how the values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], and the scale parameter, [math]\displaystyle{ \eta\,\! }[/math], affect such distribution characteristics as the shape of the curve, the reliability and the failure rate. Note that in the rest of this section we will assume the most general form of the Weibull distribution, (i.e., the 3-parameter form). The appropriate substitutions to obtain the other forms, such as the 2-parameter form where [math]\displaystyle{ \gamma = 0,\,\! }[/math] or the 1-parameter form where [math]\displaystyle{ \beta = C = \,\! }[/math] constant, can easily be made.

Effects of the Shape Parameter, beta

The Weibull shape parameter, [math]\displaystyle{ \beta\,\! }[/math], is also known as the slope. This is because the value of [math]\displaystyle{ \beta\,\! }[/math] is equal to the slope of the regressed line in a probability plot. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when [math]\displaystyle{ \beta = 1\,\! }[/math], the pdf of the 3-parameter Weibull distribution reduces to that of the 2-parameter exponential distribution or:

[math]\displaystyle{ f(t)={\frac{1}{\eta }}e^{-{\frac{t-\gamma }{\eta }}} \,\! }[/math]

where [math]\displaystyle{ \frac{1}{\eta }=\lambda = \,\! }[/math] failure rate. The parameter [math]\displaystyle{ \beta\,\! }[/math] is a pure number, (i.e., it is dimensionless). The following figure shows the effect of different values of the shape parameter, [math]\displaystyle{ \beta\,\! }[/math], on the shape of the pdf. As you can see, the shape can take on a variety of forms based on the value of [math]\displaystyle{ \beta\,\! }[/math].

For [math]\displaystyle{ 0\lt \beta \leq 1 \,\! }[/math]:

• As [math]\displaystyle{ t \rightarrow 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]), [math]\displaystyle{ f(t)\rightarrow \infty.\,\! }[/math]
• As [math]\displaystyle{ t\rightarrow \infty\,\! }[/math], [math]\displaystyle{ f(t)\rightarrow 0\,\! }[/math].
• [math]\displaystyle{ f(t)\,\! }[/math] decreases monotonically and is convex as it increases beyond the value of [math]\displaystyle{ \gamma\,\! }[/math].
• The mode is non-existent.

For [math]\displaystyle{ \beta \gt 1 \,\! }[/math]:

• [math]\displaystyle{ f(t) = 0\,\! }[/math] at [math]\displaystyle{ t = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\! }[/math]).
• [math]\displaystyle{ f(t)\,\! }[/math] increases as [math]\displaystyle{ t\rightarrow \tilde{T} \,\! }[/math] (the mode) and decreases thereafter.
• For [math]\displaystyle{ \beta \lt 2.6\,\! }[/math] the Weibull pdf is positively skewed (has a right tail), for [math]\displaystyle{ 2.6 \lt \beta \lt 3.7\,\! }[/math] its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal pdf, and for [math]\displaystyle{ \beta \gt 3.7\,\! }[/math] it is negatively skewed (left tail). The way the value of [math]\displaystyle{ \beta\,\! }[/math] relates to the physical behavior of the items being modeled becomes more apparent when we observe how its different values affect the reliability and failure rate functions. Note that for [math]\displaystyle{ \beta = 0.999\,\! }[/math], [math]\displaystyle{ f(0) = \infty\,\! }[/math], but for [math]\displaystyle{ \beta = 1.001\,\! }[/math], [math]\displaystyle{ f(0) = 0.\,\! }[/math] This abrupt shift is what complicates MLE estimation when [math]\displaystyle{ \beta\,\! }[/math] is close to 1.

The Effect of beta on the cdf and Reliability Function

The above figure shows the effect of the value of [math]\displaystyle{ \beta\,\! }[/math] on the cdf, as manifested in the Weibull probability plot. It is easy to see why this parameter is sometimes referred to as the slope. Note that the models represented by the three lines all have the same value of [math]\displaystyle{ \eta\,\! }[/math]. The following figure shows the effects of these varied values of [math]\displaystyle{ \beta\,\! }[/math] on the reliability plot, which is a linear analog of the probability plot.

• [math]\displaystyle{ R(t)\,\! }[/math] decreases sharply and monotonically for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases monotonically but less sharply than for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] and is convex.
• For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ R(t)\,\! }[/math] decreases as increases. As wear-out sets in, the curve goes through an inflection point and decreases sharply.

The Effect of beta on the Weibull Failure Rate

The value of [math]\displaystyle{ \beta\,\! }[/math] has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of [math]\displaystyle{ \beta\,\! }[/math] is less than, equal to, or greater than one.

As indicated by above figure, populations with [math]\displaystyle{ \beta \lt 1\,\! }[/math] exhibit a failure rate that decreases with time, populations with [math]\displaystyle{ \beta = 1\,\! }[/math] have a constant failure rate (consistent with the exponential distribution) and populations with [math]\displaystyle{ \beta \gt 1\,\! }[/math] have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of [math]\displaystyle{ \beta\,\! }[/math]. The Weibull failure rate for [math]\displaystyle{ 0 \lt \beta \lt 1\,\! }[/math] is unbounded at [math]\displaystyle{ T = 0\,\! }[/math] (or [math]\displaystyle{ \gamma\,\!)\,\! }[/math]. The failure rate, [math]\displaystyle{ \lambda(t),\,\! }[/math] decreases thereafter monotonically and is convex, approaching the value of zero as [math]\displaystyle{ t\rightarrow \infty\,\! }[/math] or [math]\displaystyle{ \lambda (\infty) = 0\,\! }[/math]. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For [math]\displaystyle{ \beta = 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} \,\! }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} \,\! }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For [math]\displaystyle{ \beta \gt 1\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] increases as [math]\displaystyle{ t\,\! }[/math] increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For [math]\displaystyle{ 1 \lt \beta \lt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is concave, consequently the failure rate increases at a decreasing rate as [math]\displaystyle{ t\,\! }[/math] increases.

For [math]\displaystyle{ \beta = 2\,\! }[/math] there emerges a straight line relationship between [math]\displaystyle{ \lambda(t)\,\! }[/math] and [math]\displaystyle{ t\,\! }[/math], starting at a value of [math]\displaystyle{ \lambda(t) = 0\,\! }[/math] at [math]\displaystyle{ t = \gamma\,\! }[/math], and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} \,\! }[/math]. Consequently, the failure rate increases at a constant rate as [math]\displaystyle{ t\,\! }[/math] increases. Furthermore, if [math]\displaystyle{ \eta = 1\,\! }[/math] the slope becomes equal to 2, and when [math]\displaystyle{ \gamma = 0\,\! }[/math], [math]\displaystyle{ \lambda(t)\,\! }[/math] becomes a straight line which passes through the origin with a slope of 2. Note that at [math]\displaystyle{ \beta = 2\,\! }[/math], the Weibull distribution equations reduce to that of the Rayleigh distribution.

When [math]\displaystyle{ \beta \gt 2,\,\! }[/math] the [math]\displaystyle{ \lambda(t)\,\! }[/math] curve is convex, with its slope increasing as [math]\displaystyle{ t\,\! }[/math] increases. Consequently, the failure rate increases at an increasing rate as [math]\displaystyle{ t\,\! }[/math] increases, indicating wearout life.

Effects of the Scale Parameter, eta

A change in the scale parameter [math]\displaystyle{ \eta\,\! }[/math] has the same effect on the distribution as a change of the abscissa scale. Increasing the value of [math]\displaystyle{ \eta\,\! }[/math] while holding [math]\displaystyle{ \beta\,\! }[/math] constant has the effect of stretching out the pdf. Since the area under a pdf curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of [math]\displaystyle{ \eta\,\! }[/math], as indicated in the above figure.

• If [math]\displaystyle{ \eta\,\! }[/math] is increased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If [math]\displaystyle{ \eta\,\! }[/math] is decreased while [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \gamma\,\! }[/math] are kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0 or [math]\displaystyle{ \gamma\,\! }[/math]), and its height increases.
• [math]\displaystyle{ \eta\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Effects of the Location Parameter, gamma

The location parameter, [math]\displaystyle{ \gamma\,\! }[/math], as the name implies, locates the distribution along the abscissa. Changing the value of [math]\displaystyle{ \gamma\,\! }[/math] has the effect of sliding the distribution and its associated function either to the right (if [math]\displaystyle{ \gamma \gt 0\,\! }[/math]) or to the left (if [math]\displaystyle{ \gamma \lt 0\,\! }[/math]).

• When [math]\displaystyle{ \gamma = 0,\,\! }[/math] the distribution starts at [math]\displaystyle{ t=0\,\! }[/math] or at the origin.
• If [math]\displaystyle{ \gamma \gt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the right of the origin.
• If [math]\displaystyle{ \gamma \lt 0,\,\! }[/math] the distribution starts at the location [math]\displaystyle{ \gamma\,\! }[/math] to the left of the origin.
• [math]\displaystyle{ \gamma\,\! }[/math] provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to [math]\displaystyle{ + \gamma\,\! }[/math] is a failure free operating period of such units.
• The parameter [math]\displaystyle{ \gamma\,\! }[/math] may assume all values and provides an estimate of the earliest time a failure may be observed. A negative [math]\displaystyle{ \gamma\,\! }[/math] may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• [math]\displaystyle{ \gamma\,\! }[/math] has the same units as [math]\displaystyle{ t\,\! }[/math], such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Scale Parameter, η

A change in the scale parameter η has the same effect on the distribution as a change of the abscissa scale. Increasing the value of η while holding β constant has the effect of stretching out the . Since the area under a curve is a constant value of one, the "peak" of the pdf curve will also decrease with the increase of η, as indicated in the above figure.

• If η is increased while β and γ are kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape and location.
• If η is decreased while β and γ are kept the same, the distribution gets pushed in towards the left (i.e. towards its beginning or towards 0 or γ), and its height increases.
• η has the same units as , such as hours, miles, cycles, actuations, etc.

Characteristic Effects of the Location Parameter, γ

The location parameter, γ, as the name implies, locates the distribution along the abscissa. Changing the value of γ has the effect of sliding the distribution and its associated function either to the right (if γ > 0) or to the left (if γ < 0).

• When γ = 0, the distribution starts at or at the origin.
• If γ > 0, the distribution starts at the location γ to the right of the origin.
• If γ < 0, the distribution starts at the location γ to the left of the origin.
• γ provides an estimate of the earliest time-to-failure of such units.
• The life period 0 to + γ is a failure free operating period of such units.
• The parameter γ may assume all values and provides an estimate of the earliest time a failure may be observed. A negative γ may indicate that failures have occurred prior to the beginning of the test, namely during production, in storage, in transit, during checkout prior to the start of a mission, or prior to actual use.
• γ has the same units as T, such as hours, miles, cycles, actuations, etc.

Estimation of the Weibull Parameters

The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:

The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape ([math]\displaystyle{ \beta\,\! }[/math]) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ \eta\,\! }[/math] are independent, we obtain the following posterior pdf:

[math]\displaystyle{ f(\beta ,\eta |Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\! }[/math]

In this model, [math]\displaystyle{ \eta\,\! }[/math] is assumed to follow a noninformative prior distribution with the density function [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta } \,\! }[/math]. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on [math]\displaystyle{ \eta\,\! }[/math]. Specifically, since [math]\displaystyle{ \eta\,\! }[/math] is always positive, we can assume that ln([math]\displaystyle{ \eta\,\! }[/math]) follows a uniform distribution, [math]\displaystyle{ U( -\infty, +\infty \,\! }[/math]). Applying Jeffrey's rule as given in Gelman et al. [9] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields [math]\displaystyle{ \varphi (\eta )=\dfrac{1}{\eta }\,\! }[/math].

The prior distribution of [math]\displaystyle{ \beta\,\! }[/math], denoted as [math]\displaystyle{ \varphi (\beta )\,\! }[/math], can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

• Collect the times-to-failure data.
• Specify a prior distribution for [math]\displaystyle{ \beta\,\! }[/math] (the prior for [math]\displaystyle{ \eta\,\! }[/math] is assumed to be [math]\displaystyle{ 1/\beta\,\! }[/math]).
• Obtain the posterior pdf from the above equation.

In other words, a distribution (the posterior pdf) is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the Analysis page, under the Results As area, as shown next.

The expected value of [math]\displaystyle{ \beta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\beta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\beta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

Similarly, the expected value of [math]\displaystyle{ \eta\,\! }[/math] is obtained by:

[math]\displaystyle{ E(\eta )=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\eta \cdot f(\beta ,\eta |Data)d\beta d\eta \,\! }[/math]

The median points are obtained by solving the following equations for [math]\displaystyle{ \breve{\beta} \,\! }[/math] and [math]\displaystyle{ \breve{\eta} \,\! }[/math] respectively:

[math]\displaystyle{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\breve{\beta}}f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{\breve{\eta}}\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d\beta d\eta =0.5 \,\! }[/math]

Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution (w.r.t. one of the parameters). The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0.5 the percentage of interest is given. This procedure actually provides the confidence bounds on the parameters, which in the Bayesian framework are called ‘‘Credible Bounds.‘‘ However, since the engineering interpretation is the same, and to avoid confusion, we refer to them as confidence bounds in this reference and in Weibull++.

Posterior Distributions for Functions of Parameters

As explained in Parameter Estimation, in Bayesian analysis, all the functions of the parameters are distributed. In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics. Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated. Again, the expected value (mean) or median value are used. It is important to note that the Median value is preferable and is the default in Weibull++. This is because the Median value always corresponds to the 50th percentile of the distribution. On the other hand, the Mean is not a fixed point on the distribution, which could cause issues, especially when comparing results across different data sets.

pdf of the Times-to-Failure

The posterior distribution of the failure time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ f(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }f(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ f(T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1}e^{-\left( \dfrac{T}{\eta }\right) ^{\beta }} \,\! }[/math]

For the pdf of the times-to-failure, only the expected value is calculated and reported in Weibull++.

Reliability

In order to calculate the median value of the reliability function, we first need to obtain posterior pdf of the reliability. Since [math]\displaystyle{ R(T)\,\! }[/math] is a function of [math]\displaystyle{ \beta\,\! }[/math], the density functions of [math]\displaystyle{ \beta\,\! }[/math] and [math]\displaystyle{ R(T)\,\! }[/math] have the following relationship:

[math]\displaystyle{ \begin{align} f(R|Data,T)dR = & f(\beta |Data)d\beta)\\ = & (\int\nolimits_{0}^{\infty }f(\beta ,\eta |Data)d{\eta}) d{\beta} \\ =& \dfrac{\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }d\beta \end{align}\,\! }[/math]

The median value of the reliability is obtained by solving the following equation w.r.t. [math]\displaystyle{ \breve{R}: \,\! }[/math]

[math]\displaystyle{ \int\nolimits_{0}^{\breve{R}}f(R|Data,T)dR=0.5 \,\! }[/math]

The expected value of the reliability at time [math]\displaystyle{ t\,\! }[/math] is given by:

[math]\displaystyle{ R(T|Data)=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R(T,\beta ,\eta )f(\beta ,\eta |Data)d\eta d\beta \,\! }[/math]

where:

[math]\displaystyle{ R(T,\beta ,\eta )=e^{-\left( \dfrac{T}{\eta }\right) ^{^{\beta }}} \,\! }[/math]

Failure Rate

The failure rate at time is given by:

[math]\displaystyle{ \lambda (T|Data)=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }\lambda (T,\beta ,\eta )L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\eta )\varphi (\beta )d\eta d\beta } \,\! }[/math]

where:

[math]\displaystyle{ \lambda (T,\beta ,\eta )=\dfrac{\beta }{\eta }\left( \dfrac{T}{\eta }\right) ^{\beta -1} \,\! }[/math]

Bounds on Reliability for Bayesian-Weibull

The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of [math]\displaystyle{ \beta\,\! }[/math] is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ R(T)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(T)}f(R|Data,t)dR=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ \ R(t) \,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,t)dR=1-CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math]. The Bayesian two-sided bounds estimate for [math]\displaystyle{ R(t)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R|Data,t)dR=CL \,\! }[/math] which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{R_{U}(t)}f(R|Data,t)dR=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{R_{L}(t)}f(R|Data,T)dR=(1-CL)/2 \,\! }[/math]

Using the same method for one-sided bounds, [math]\displaystyle{ {{R}_{U}}(t)\,\! }[/math] and [math]\displaystyle{ {{R}_{L}}(t)\,\! }[/math] can be computed.

Bounds on Time for Bayesian-Weibull

Following the same procedure described for bounds on Reliability, the bounds of time [math]\displaystyle{ t\,\! }[/math] can be calculated, given [math]\displaystyle{ R\,\! }[/math]. The Bayesian one-sided upper bound estimate for [math]\displaystyle{ t(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

Using the posterior distribution, the following is obtained:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{U}}(R)\,\! }[/math]. The Bayesian one-sided lower bound estimate for [math]\displaystyle{ T(R)\,\! }[/math] is given by:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=1-CL \,\! }[/math]

or:

[math]\displaystyle{ \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\! }[/math]

The above equation can be solved for [math]\displaystyle{ {{T}_{L}}(R)\,\! }[/math]. The Bayesian two-sided lower bounds estimate for [math]\displaystyle{ T(R)\,\! }[/math] is:

[math]\displaystyle{ \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T|Data,R)dT=CL \,\! }[/math]

which is equivalent to:

[math]\displaystyle{ \int\nolimits_{0}^{T_{U}(R)}f(T|Data,R)dT=(1+CL)/2 \,\! }[/math]

and:

[math]\displaystyle{ \int\nolimits_{0}^{T_{L}(R)}f(T|Data,R)dT=(1-CL)/2 \,\! }[/math]

Bayesian-Weibull Example

A manufacturer has tested prototypes of a modified product. The test was terminated at 2,000 hours, with only 2 failures observed from a sample size of 18. The following table shows the data.

Number of State	State of F or S	State End Time
1	F	1180
1	F	1842
16	S	2000

Because of the lack of failure data in the prototype testing, the manufacturer decided to use information gathered from prior tests on this product to increase the confidence in the results of the prototype testing. This decision was made because failure analysis indicated that the failure mode of the two failures is the same as the one that was observed in previous tests. In other words, it is expected that the shape of the distribution (beta) hasn't changed, but hopefully the scale (eta) has, indicating longer life. The 2-parameter Weibull distribution was used to model all prior tests results. The estimated beta ([math]\displaystyle{ \beta\,\! }[/math]) parameters of the prior test results are as follows:

Betas Obtained for Similar Mode
1.7
2.1
2.4
3.1
3.5

Solution

First, in order to fit the data to a Bayesian-Weibull model, a prior distribution for beta needs to be determined. Based on the beta values in the prior tests, the prior distribution for beta is found to be a lognormal distribution with [math]\displaystyle{ \mu = 0.9064\,\! }[/math], [math]\displaystyle{ \sigma = 0.3325\,\! }[/math]. (The values of the parameters can be obtained by entering the beta values into a Weibull++ standard folio and analyzing it using the lognormal distribution and the RRX analysis method.)

Next, enter the data from the prototype testing into a standard folio. On the control panel, choose the Bayesian-Weibull > B-W Lognormal Prior distribution. Click Calculate and enter the parameters of the lognormal distribution, as shown next.

Click OK. The result is Beta (Median) = 2.361219 and Eta (Median) = 5321.631912 (by default Weibull++ returns the median values of the posterior distribution). Suppose that the reliability at 3,000 hours is the metric of interest in this example. Using the QCP, the reliability is calculated to be 76.97% at 3,000 hours. The following picture depicts the posterior pdf plot of the reliability at 3,000, with the corresponding median value as well as the 10th percentile value. The 10th percentile constitutes the 90% lower 1-sided bound on the reliability at 3,000 hours, which is calculated to be 50.77%.

The pdf of the times-to-failure data can be plotted in Weibull++, as shown next:

Median Rank Plot Example

In this example, we will determine the median rank value used for plotting the 6th failure from a sample size of 10. This example will use Weibull++'s Quick Statistical Reference (QSR) tool to show how the points in the plot of the following example are calculated.

First, open the Quick Statistical Reference tool and select the Inverse F-Distribution Values option.

In this example, n1 = 10, j = 6, m = 2(10 - 6 + 1) = 10, and n2 = 2 x 6 = 12.

Thus, from the F-distribution rank equation:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\! }[/math]

Use the QSR to calculate the value of F_0.5;10;12 = 0.9886, as shown next:

Consequently:

[math]\displaystyle{ MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\! }[/math]

Another method is to use the Median Ranks option directly, which yields MR(%) = 54.8305%, as shown next:

Complete Data Example

Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, [math]\displaystyle{ {T}_{j}\,\! }[/math]: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull pdf that represents these data.

Solution

Create a new Weibull++ standard folio that is configured for grouped times-to-failure data with suspensions.

Enter the data in the appropriate columns. Note that there are 4 suspensions, as only 6 of the 10 units were tested to failure (the next figure shows the data as entered). Use the 3-parameter Weibull and MLE for the calculations.

Plot the data.

Note that the original data points, on the curved line, were adjusted by subtracting 30.92 hours to yield a straight line as shown above.

Suspension Data Example

ACME company manufactures widgets, and it is currently engaged in reliability testing a new widget design. 19 units are being reliability tested, but due to the tremendous demand for widgets, units are removed from the test whenever the production cannot cover the demand. The test is terminated at the 67th day when the last widget is removed from the test. The following table contains the collected data.

Widget Test Data

Data Point Index	State (F/S)	Time to Failure
1	F	2
2	S	3
3	F	5
4	S	7
5	F	11
6	S	13
7	S	17
8	S	19
9	F	23
10	F	29
11	S	31
12	F	37
13	S	41
14	F	43
15	S	47
16	S	53
17	F	59
18	S	61
19	S	67

Solution

In this example, we see that the number of failures is less than the number of suspensions. This is a very common situation, since reliability tests are often terminated before all units fail due to financial or time constraints. Furthermore, some suspensions will be recorded when a failure occurs that is not due to a legitimate failure mode, such as operator error. In cases such as this, a suspension is recorded, since the unit under test cannot be said to have had a legitimate failure.

Enter the data into a Weibull++ standard folio that is configured for times-to-failure data with suspensions. The folio will appear as shown next:

We will use the 2-parameter Weibull to solve this problem. The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=1.145 \\ & \hat{\eta }=65.97 \\ \end{align}\,\! }[/math]

Using RRX:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.914\\ & \hat{\eta }=79.38 \\ \end{align}\,\! }[/math]

Using RRY:

[math]\displaystyle{ \begin{align} & \hat{\beta }=0.895\\ & \hat{\eta }=82.02 \\ \end{align}\,\! }[/math]

Interval Data Example

Suppose we have run an experiment with 8 units tested and the following is a table of their last inspection times and failure times:

Data Point Index	Last Inspection	Failure Time
1	30	32
2	32	35
3	35	37
4	37	40
5	42	42
6	45	45
7	50	50
8	55	55

Analyze the data using several different parameter estimation techniques and compare the results.

Solution

Enter the data into a Weibull++ standard folio that is configured for interval data. The data is entered as follows:

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.76 \\ & \hat{\eta }=44.68 \\ \end{align}\,\! }[/math]

Using RRX or rank regression on X:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.70 \\ & \hat{\eta }=44.54 \\ \end{align}\,\! }[/math]

Using RRY or rank regression on Y:

[math]\displaystyle{ \begin{align} & \hat{\beta }=5.41 \\ & \hat{\eta }=44.76 \\ \end{align}\,\! }[/math]

The plot of the MLE solution with the two-sided 90% confidence bounds is:

Mixed Data Types Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406. [20].

Estimate the parameters for the 3-parameter Weibull, for a sample of 10 units that are all tested to failure. The recorded failure times are 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160 and 1,400 hours.

Published Results:

Published results (using probability plotting):

[math]\displaystyle{ {\widehat{\beta}} = 3.0\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1,220\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -300\,\! }[/math]

Computed Results in Weibull++

Weibull++ computed parameters for rank regression on X are:

[math]\displaystyle{ {\widehat{\beta}} = 2.9013\,\! }[/math], [math]\displaystyle{ {\widehat{\eta}} = 1195.5009\,\! }[/math], [math]\displaystyle{ {\widehat{\gamma}} = -279.000\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Weibull Distribution RRX Example

Assume that 6 identical units are being tested. The failure times are: 93, 34, 16, 120, 53 and 75 hours.

1. What is the unreliability of the units for a mission duration of 30 hours, starting the mission at age zero?

2. What is the reliability for a mission duration of 10 hours, starting the new mission at the age of T = 30 hours?

3. What is the longest mission that this product should undertake for a reliability of 90%?

Solution

1. First, we use Weibull++ to obtain the parameters using RRX.

Then, we investigate several methods of solution for this problem. The first, and more laborious, method is to extract the information directly from the plot. You may do this with either the screen plot in RS Draw or the printed copy of the plot. (When extracting information from the screen plot in RS Draw, note that the translated axis position of your mouse is always shown on the bottom right corner.)

Using this first method, enter either the screen plot or the printed plot with T = 30 hours, go up vertically to the straight line fitted to the data, then go horizontally to the ordinate, and read off the result. A good estimate of the unreliability is 23%. (Also, the reliability estimate is 1.0 - 0.23 = 0.77 or 77%.)

The second method involves the use of the Quick Calculation Pad (QCP).

Select the Prob. of Failure calculation option and enter 30 hours in the Mission End Time field.

Note that the results in QCP vary according to the parameter estimation method used. The above results are obtained using RRX.

2. The conditional reliability is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}\,\! }[/math]

or:

[math]\displaystyle{ \hat{R}(10hr|30hr)=\frac{\hat{R}(10+30)}{\hat{R}(30)}=\frac{\hat{R}(40)}{\hat{R}(30)}\,\! }[/math]

Again, the QCP can provide this result directly and more accurately than the plot.

3. To use the QCP to solve for the longest mission that this product should undertake for a reliability of 90%, choose Reliable Life and enter 0.9 for the required reliability. The result is 15.9933 hours.

Benchmark with Published Examples

The following examples compare published results to computed results obtained with Weibull++.

Complete Data RRY Example

From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].

Sample of 10 units, all tested to failure. The failures were recorded at 16, 34, 53, 75, 93, 120, 150, 191, 240 and 339 hours.

Published Results

Published Results (using Rank Regression on Y):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.20 \\ & \widehat{\eta} = 146.2 \\ & \hat{\rho }=0.998703\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard data sheet. Use RRY for the estimation method.

Weibull++ computed parameters for RRY are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.1973 \\ & \widehat{\eta} = 146.2545 \\ & \hat{\rho }=0.9999\\ \end{align}\,\! }[/math]

The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).

You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.

Suspension Data MLE Example

From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].

70 diesel engine fans accumulated 344,440 hours in service and 12 of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the 2-parameter Weibull distribution.

Published Results:

Weibull parameters (2P-Weibull, MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,296 \\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Note that Nelson expresses the results as multiples of 1,000 (or = 26.297, etc.). The published results were adjusted by this factor to correlate with Weibull++ results.

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio, using 2-parameter Weibull and MLE to calculate the parameter estimates.

You can also enter the data as given in table without grouping them by opening a data sheet configured for suspension data. Then click the Group Data icon and chose Group exactly identical values.

The data will be automatically grouped and put into a new grouped data sheet.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.0584 \\ & \widehat{\eta} = 26,297 \\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\ & \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

The two-sided 95% bounds on the parameters can be determined from the QCP. Calculate and then click Report to see the results.

Interval Data MLE Example

From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. 167 identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:

Published Results:

Published results (using MLE):

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.486 \\ & \widehat{\eta} = 71.687\\ \end{align}\,\! }[/math]

Published 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\ \end{align}\,\! }[/math]

Published variance/covariance matrix:

Computed Results in Weibull++

This same data set can be entered into a Weibull++ standard folio that's configured for grouped times-to-failure data with suspensions and interval data.

Weibull++ computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=1.485 \\ & \widehat{\eta} = 71.690\\ \end{align}\,\! }[/math]

Weibull++ computed 95% FM confidence limits on the parameters:

[math]\displaystyle{ \begin{align} & \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\ & \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\ \end{align}\,\! }[/math]

Weibull++ computed/variance covariance matrix:

Grouped Suspension MLE Example

From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100.

Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.

Published Results (using MLE)

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Computed Results in Weibull++

[math]\displaystyle{ \begin{align} & \widehat{\beta }=3.7596935\\ & \widehat{\eta} = 106.49758 \\ & \hat{\gamma }=14.451684\\ \end{align}\,\! }[/math]

Note that you must select the Use True 3-P MLEoption in the Weibull++ Application Setup to replicate these results.

3-P Probability Plot Example

Suppose we want to model a left censored, right censored, interval, and complete data set, consisting of 274 units under test of which 185 units fail. The following table contains the data.

The Test Data

Data Point Index	Number in State	Last Inspection	State (S or F)	State End Time
1	2	5	F	5
2	23	5	S	5
3	28	0	F	7
4	4	10	F	10
5	7	15	F	15
6	8	20	F	20
7	29	20	S	20
8	32	0	F	22
9	6	25	F	25
10	4	27	F	30
11	8	30	F	35
12	5	30	F	40
13	9	27	F	45
14	7	25	F	50
15	5	20	F	55
16	3	15	F	60
17	6	10	F	65
18	3	5	F	70
19	37	100	S	100
20	48	0	F	102

Solution

Since standard ranking methods for dealing with these different data types are inadequate, we will want to use the ReliaSoft ranking method. This option is the default in Weibull++ when dealing with interval data. The filled-out standard folio is shown next:

The computed parameters using MLE are:

[math]\displaystyle{ \hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\! }[/math]

Using RRX:

[math]\displaystyle{ \hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\! }[/math]

Using RRY:

[math]\displaystyle{ \hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\! }[/math]

The plot with the two-sided 90% confidence bounds for the rank regression on X solution is: