The Generalized Gamma Distribution

Generalized Gamma Probability Density Function

The generalized gamma function is a three-parameter distribution. One version of the generalized gamma distribution uses the parameters [math]\displaystyle{ k }[/math], [math]\displaystyle{ \beta }[/math], and [math]\displaystyle{ \theta }[/math]. The [math]\displaystyle{ pdf }[/math] for this form of the generalized gamma distribution is given by:

[math]\displaystyle{ f(t)=\frac{\beta }{\Gamma (k)\cdot \theta }{{\left( \frac{t}{\theta } \right)}^{k\beta -1}}{{e}^{-{{\left( \tfrac{t}{\theta } \right)}^{\beta }}}} }[/math]

where [math]\displaystyle{ \theta \gt 0 }[/math] is a scale parameter, [math]\displaystyle{ \beta \gt 0 }[/math] and [math]\displaystyle{ k\gt 0 }[/math] are shape parameters and [math]\displaystyle{ \Gamma (x) }[/math] is the gamma function of [math]\displaystyle{ x }[/math], which is defined by:

[math]\displaystyle{ \Gamma (x)=\int_{0}^{\infty }{{s}^{x-1}}\cdot {{e}^{-s}}ds }[/math]

With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of [math]\displaystyle{ k }[/math], [math]\displaystyle{ \beta }[/math], and [math]\displaystyle{ \theta }[/math] may appear almost identical [21]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters [math]\displaystyle{ \mu }[/math] , [math]\displaystyle{ \sigma }[/math] , and [math]\displaystyle{ \lambda }[/math] [21] where:

[math]\displaystyle{ \begin{align} \mu = & ln(\theta )+\frac{1}{\beta }\cdot ln\left( \frac{1}{{{\lambda }^{2}}} \right) \\ \sigma = & \frac{1}{\beta \sqrt{k}} \\ \lambda = & \frac{1}{\sqrt{k}} \end{align} }[/math]

where [math]\displaystyle{ -\infty \lt \mu \lt \infty ,\,\sigma \gt 0, }[/math] and [math]\displaystyle{ 0\lt \lambda . }[/math] While this makes the distribution converge much more easily in computations, it does not facilitate manual manipulation of the equation. By allowing [math]\displaystyle{ \lambda }[/math] to become negative, the [math]\displaystyle{ pdf }[/math] of the reparameterized distribution is given by:

[math]\displaystyle{ f(t)=\left\{ \begin{matrix} \tfrac{|\lambda |}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma \left( \tfrac{1}{{{\lambda }^{2}}} \right)}\cdot {{e}^{\left[ \tfrac{\lambda \cdot \tfrac{\text{ln}(t)-\mu }{\sigma }+\text{ln}\left( \tfrac{1}{{{\lambda }^{2}}} \right)-{{e}^{\lambda \cdot \tfrac{\text{ln}(t)-\mu }{\sigma }}}}{{{\lambda }^{2}}} \right]}}\text{ if }\lambda \ne 0 \\ \tfrac{1}{t\cdot \sigma \sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}^{2}}}}\text{ if }\lambda =0 \\ \end{matrix} \right. }[/math]

Generalized Gamma Reliability Function

The reliability function for the generalized gamma distribution is given by:

[math]\displaystyle{ }[/math]

[math]\displaystyle{ R(t)=\left\{ \begin{array}{*{35}{l}} 1-{{\Gamma }_{I}}\left( \tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}};\tfrac{1}{{{\lambda }^{2}}} \right)\text{ if }\lambda \gt 0 \\ 1-\Phi \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)\text{ if }\lambda =0 \\ {{\Gamma }_{I}}\left( \tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}};\tfrac{1}{{{\lambda }^{2}}} \right)\text{ if }\lambda \lt 0 \\ \end{array} \right. }[/math]

where:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}{{e}^{-\tfrac{{{x}^{2}}}{2}}}dx }[/math]

and [math]\displaystyle{ {{\Gamma }_{I}}(k;x) }[/math] is the incomplete gamma function of [math]\displaystyle{ k }[/math] and [math]\displaystyle{ x }[/math] , which is given by:

[math]\displaystyle{ {{\Gamma }_{I}}(k;x)=\frac{1}{\Gamma (k)}\int_{0}^{x}{{s}^{k-1}}{{e}^{-s}}ds }[/math]

where [math]\displaystyle{ \Gamma (x) }[/math] is the gamma function of [math]\displaystyle{ x }[/math] . Note that in Weibull++ the probability plot of the generalized gamma is created on lognormal probability paper. This means that the fitted line will not be straight unless [math]\displaystyle{ \lambda =0. }[/math]

Generalized Gamma Failure Rate Function

As defined in Chapter 3, the failure rate function is given by:

[math]\displaystyle{ \lambda (t)=\frac{f(t)}{R(t)} }[/math]

Owing to the complexity of the equations involved, the function will not be displayed here, but the failure rate function for the generalized gamma distribution can be obtained merely by dividing Eqn. (ggampdf) by Eqn. (ggamrel).

Generalized Gamma Reliable Life

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

[math]\displaystyle{ {{T}_{R}}=\left\{ \begin{array}{*{35}{l}} {{e}^{\mu +\tfrac{\sigma }{\lambda }\ln \left[ {{\lambda }^{2}}\Gamma _{I}^{-1}\left( 1-R,\tfrac{1}{{{\lambda }^{2}}} \right) \right]}}\text{ if }\lambda \gt 0 \\ {{\Phi }^{-1}}(1-R)\text{ if }\lambda =0 \\ {{e}^{\mu +\tfrac{\sigma }{\lambda }\ln \left[ {{\lambda }^{2}}\Gamma _{I}^{-1}\left( R,\tfrac{1}{{{\lambda }^{2}}} \right) \right]}}\text{ if }\lambda \lt 0 \\ \end{array} \right. }[/math]

Characteristics of the Generalized Gamma Distribution

As mentioned previously, the generalized gamma distribution includes other distributions as special cases based on the values of the parameters.

• The Weibull distribution is a special case when [math]\displaystyle{ \lambda =1 }[/math] and:

[math]\displaystyle{ \begin{align} & \beta = & \frac{1}{\sigma } \\ & \eta = & \ln (\mu ) \end{align} }[/math]

• In this case, the generalized distribution has the same behavior as the Weibull for [math]\displaystyle{ \sigma \gt 1, }[/math] [math]\displaystyle{ \sigma =1, }[/math] and [math]\displaystyle{ \sigma \lt 1 }[/math] ( [math]\displaystyle{ \beta \lt 1, }[/math] [math]\displaystyle{ \beta =1, }[/math] and [math]\displaystyle{ \beta \gt 1 }[/math] respectively).

• The exponential distribution is a special case when [math]\displaystyle{ \lambda =1 }[/math] and [math]\displaystyle{ \sigma =1 }[/math].

• The lognormal distribution is a special case when [math]\displaystyle{ \lambda =0 }[/math].

• The gamma distribution is a special case when [math]\displaystyle{ \lambda =\sigma }[/math].

By allowing [math]\displaystyle{ \lambda }[/math] to take negative values, the generalized gamma distribution can be further extended to include additional distributions as special cases. For example, the Fréchet distribution of maxima (also known as a reciprocal Weibull) is a special case when [math]\displaystyle{ \lambda =-1 }[/math].

Confidence Bounds

The only method available in Weibull++ for confidence bounds for the generalized gamma distribution is the Fisher matrix, which is described next.

Bounds on the Parameters

The lower and upper bounds on the parameter [math]\displaystyle{ \mu }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\ & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)} \end{align} }[/math]

For the parameter [math]\displaystyle{ \widehat{\sigma } }[/math] , [math]\displaystyle{ \ln (\widehat{\sigma }) }[/math] is treated as normally distributed, and the bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & \widehat{\sigma }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}\text{ (upper bound)} \\ & {{\sigma }_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}}\text{ (lower bound)} \end{align} }[/math]

For the parameter [math]\displaystyle{ \lambda , }[/math] the bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{\lambda }_{U}}= & \widehat{\lambda }+{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\text{ (upper bound)} \\ & {{\lambda }_{L}}= & \widehat{\lambda }-{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\text{ (lower bound)} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds.

The variances and covariances of [math]\displaystyle{ \widehat{\mu } }[/math] and [math]\displaystyle{ \widehat{\sigma } }[/math] are estimated as follows:

[math]\displaystyle{ \begin{align} & & \left( \begin{matrix} \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\lambda } \right) \\ \widehat{Cov}\left( \widehat{\sigma },\widehat{\mu } \right) & \widehat{Var}\left( \widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\sigma },\widehat{\lambda } \right) \\ \widehat{Cov}\left( \widehat{\lambda },\widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\lambda },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\lambda } \right) \\ \end{matrix} \right) \\ & = & \left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \lambda } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \lambda } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \\ \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma },\lambda =\hat{\lambda }}^{-1} \end{align} }[/math]

Where [math]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the generalized gamma distribution.

Bounds on Reliability

The upper and lower bounds on reliability are given by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}} \\ & {{R}_{L}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}} \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & Var(\widehat{R})= & {{\left( \frac{\partial R}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial R}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+ \\ & & +2\left( \frac{\partial R}{\partial \mu } \right)\left( \frac{\partial R}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial R}{\partial \mu } \right)\left( \frac{\partial R}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })+ \\ & & +2\left( \frac{\partial R}{\partial \lambda } \right)\left( \frac{\partial R}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma }) \end{align} }[/math]

Bounds on Time

The bounds around time for a given percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, given by Eqn. (GGamma Time). Since [math]\displaystyle{ T }[/math] is a positive variable, the transformed variable [math]\displaystyle{ \hat{u}=\ln (\widehat{T}) }[/math] is treated as normally distributed and the bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{u}_{u}}= & \ln {{T}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} \\ & {{u}_{L}}= & \ln {{T}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} \end{align} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] we get:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{T}_{U}}}}\text{ (upper bound)} \\ & {{T}_{L}}= & {{e}^{{{T}_{L}}}}\text{ (lower bound)} \end{align} }[/math]

The variance of [math]\displaystyle{ u }[/math] is estimated from:

[math]\displaystyle{ \begin{align} & Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+ \\ & & +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })+ \\ & & +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma }) \end{align} }[/math]

A Generalized Gamma Distribution Example

The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [21].

[math]\displaystyle{ \begin{array}{*{35}{l}} \text{17}\text{.88} & \text{28}\text{.92} & \text{33} & \text{41}\text{.52} & \text{42}\text{.12} & \text{45}\text{.6} & \text{48}\text{.4} & \text{51}\text{.84} & \text{51}\text{.96} & \text{54}\text{.12} \\ \text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84} \\ \text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\ \end{array} }[/math]

When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 4.23064 \\ & \widehat{\sigma }= & 0.509982 \\ & \widehat{\lambda }= & 0.307639 \end{align} }[/math]

Note that for this data, the generalized gamma offers a compromise between the Weibull [math]\displaystyle{ (\lambda =1), }[/math] and the lognormal [math]\displaystyle{ (\lambda =0) }[/math] distributions. The value of [math]\displaystyle{ \lambda }[/math] indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on [math]\displaystyle{ \lambda . }[/math] For example, the 90% two-sided confidence bounds are:

[math]\displaystyle{ \begin{align} & {{\lambda }_{u}}= & -0.592087 \\ & {{\lambda }_{u}}= & 1.20736 \end{align} }[/math]

It can be then concluded that both distributions (i.e. Weibull and lognormal) are well supported by the data, with the lognormal being the ,better supported of the two. In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next.

It is important to also note that as in the case of the mixed Weibull distribution, in the case of regression analysis, using a generalized gamma model, the choice of regression axis, i.e. [math]\displaystyle{ RRX }[/math] or [math]\displaystyle{ RRY, }[/math] is of no consequence since non-linear regression is utilized.

The Gamma Distribution

The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications. [32]

Gamma Probability Density Function

The [math]\displaystyle{ pdf }[/math] of the gamma distribution is given by:

[math]\displaystyle{ f(T)=\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)} }[/math]

where:

[math]\displaystyle{ z=\ln (t)-\mu }[/math]

and:

[math]\displaystyle{ \begin{align} & {{e}^{\mu }}= & \text{scale parameter} \\ & k= & \text{shape parameter} \end{align} }[/math]

where [math]\displaystyle{ 0\lt t\lt \infty }[/math] , [math]\displaystyle{ -\infty \lt \mu \lt \infty }[/math] and [math]\displaystyle{ k\gt 0 }[/math] . The Gamma Reliability Function The reliability for a mission of time [math]\displaystyle{ T }[/math] for the gamma distribution is:

[math]\displaystyle{ R=1-{{\Gamma }_{1}}(k;{{e}^{z}}) }[/math]

The Gamma Mean, Median and Mode

The gamma mean or MTTF is:

[math]\displaystyle{ \overline{T}=k{{e}^{\mu }} }[/math]

The mode exists if [math]\displaystyle{ k\gt 1 }[/math] and is given by:

[math]\displaystyle{ \tilde{T}=(k-1){{e}^{\mu }} }[/math]

The median is:

[math]\displaystyle{ \widehat{T}={{e}^{\mu +\ln (\Gamma _{1}^{-1}(0.5;k))}} }[/math]

The Gamma Standard Deviation

The standard deviation for the gamma distribution is:

[math]\displaystyle{ {{\sigma }_{T}}=\sqrt{k}{{e}^{\mu }} }[/math]

The Gamma Reliable Life

The gamma reliable life is:

[math]\displaystyle{ {{T}_{R}}={{e}^{\mu +\ln (\Gamma _{1}^{-1}(1-R;k))}} }[/math]

The Gamma Failure Rate Function

The instantaneous gamma failure rate is given by:

[math]\displaystyle{ \lambda =\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)(1-{{\Gamma }_{1}}(k;{{e}^{z}}))} }[/math]

Characteristics of the Gamma Distribution

Some of the specific characteristics of the gamma distribution are the following:

For [math]\displaystyle{ k\gt 1 }[/math] :

• As [math]\displaystyle{ T\to 0,\infty }[/math] , [math]\displaystyle{ f(T)\to 0. }[/math]

• [math]\displaystyle{ f(T) }[/math] increases from 0 to the mode value and decreases thereafter.

• If [math]\displaystyle{ k\le 2 }[/math] then [math]\displaystyle{ pdf }[/math] has one inflection point at [math]\displaystyle{ T={{e}^{\mu }}\sqrt{k-1}( }[/math] [math]\displaystyle{ \sqrt{k-1}+1). }[/math]

• If [math]\displaystyle{ k\gt 2 }[/math] then [math]\displaystyle{ pdf }[/math] has two inflection points for [math]\displaystyle{ T={{e}^{\mu }}\sqrt{k-1}( }[/math] [math]\displaystyle{ \sqrt{k-1}\pm 1). }[/math]

• For a fixed [math]\displaystyle{ k }[/math] , as [math]\displaystyle{ \mu }[/math] increases, the [math]\displaystyle{ pdf }[/math] starts to look more like a straight angle.

As [math]\displaystyle{ T\to \infty ,\lambda (T)\to \tfrac{1}{{{e}^{\mu }}}. }[/math]

For [math]\displaystyle{ k=1 }[/math] :

• Gamma becomes the exponential distribution.

• As [math]\displaystyle{ T\to 0 }[/math] , [math]\displaystyle{ f(T)\to \tfrac{1}{{{e}^{\mu }}}. }[/math]

• As [math]\displaystyle{ T\to \infty ,f(T)\to 0. }[/math]

• The [math]\displaystyle{ pdf }[/math] decreases monotonically and is convex.

• [math]\displaystyle{ \lambda (T)\equiv \tfrac{1}{{{e}^{\mu }}} }[/math] . [math]\displaystyle{ \lambda (T) }[/math] is constant.

• The mode does not exist.

For [math]\displaystyle{ 0\lt k\lt 1 }[/math] :

• As [math]\displaystyle{ T\to 0 }[/math] , [math]\displaystyle{ f(T)\to \infty . }[/math]

• As [math]\displaystyle{ T\to \infty ,f(T)\to 0. }[/math]

• As [math]\displaystyle{ T\to \infty ,\lambda (T)\to \tfrac{1}{{{e}^{\mu }}}. }[/math]

• The [math]\displaystyle{ pdf }[/math] decreases monotonically and is convex.

• As [math]\displaystyle{ \mu }[/math] increases, the [math]\displaystyle{ pdf }[/math] gets stretched out to the right and its height decreases, while maintaining its shape.

• As [math]\displaystyle{ \mu }[/math] decreases, the [math]\displaystyle{ pdf }[/math] shifts towards the left and its height increases.

• The mode does not exist.

Confidence Bounds

The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in Chapter 5.

Bounds on the Parameters

The lower and upper bounds on the mean, [math]\displaystyle{ \widehat{\mu } }[/math] , are estimated from:

[math]\displaystyle{ \begin{align} & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\ & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ \widehat{\sigma } }[/math] , must be positive, [math]\displaystyle{ \ln (\widehat{\sigma }) }[/math] is treated as normally distributed and the bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{{\hat{k}}}}}\text{ (upper bound)} \\ & {{k}_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{\widehat{k}}}}}\text{ (lower bound)} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds.

The variances and covariances of [math]\displaystyle{ \widehat{\mu } }[/math] and [math]\displaystyle{ \widehat{k} }[/math] are estimated from the Fisher matrix, as follows:

[math]\displaystyle{ \left( \begin{matrix} \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) \\ \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) & \widehat{Var}\left( \widehat{k} \right) \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} \\ {} & {} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}} \\ \end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1} }[/math]

[math]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the gamma distribution, described in Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\widehat{k};{{e}^{\widehat{z}}}) }[/math]

where:

[math]\displaystyle{ \widehat{z}=\ln (t)-\widehat{\mu } }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ {{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (upper bound)} }[/math]

[math]\displaystyle{ {{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (lower bound)} }[/math]

where:

[math]\displaystyle{ Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k}) }[/math]

Bounds on Time

The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]

where:

[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]

[math]\displaystyle{ Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma }) }[/math]

or:

[math]\displaystyle{ Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma }) }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\ & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)} \end{align} }[/math]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \begin{matrix} \text{61} & \text{50} & \text{67} & \text{49} & \text{53} & \text{62} \\ \text{53} & \text{61} & \text{43} & \text{65} & \text{53} & \text{56} \\ \text{62} & \text{56} & \text{58} & \text{55} & \text{58} & \text{48} \\ \text{66} & \text{44} & \text{48} & \text{58} & \text{43} & \text{40} \\ \end{matrix} }[/math]

Fitting the gamma distribution to this data, using maximum likelihood as the analysis method, gives the following parameters:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 7.72E-02 \\ & \hat{k}= & 50.4908 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ X, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

Using rank regression on [math]\displaystyle{ Y, }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 0.2915 \\ & \hat{k}= & 41.1726 \end{align} }[/math]

The Logistic Distribution

The logistic distribution has been used for growth models, and is used in a certain type of regression known as the logistic regression. It has also applications in modeling life data. The shape of the logistic distribution and the normal distribution are very similar [27]. There are some who argue that the logistic distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small location parameter, the issue of negative failure times should not present itself as a problem.

Logistic Probability Density Function

The logistic [math]\displaystyle{ pdf }[/math] is given by:

[math]\displaystyle{ \begin{matrix} f(T)=\tfrac{{{e}^{z}}}{\sigma {{(1+{{e}^{z}})}^{2}}} \\ z=\tfrac{t-\mu }{\sigma } \\ -\infty \lt T\lt \infty ,\ \ -\infty \lt \mu \lt \infty ,\sigma \gt 0 \\ \end{matrix} }[/math]

where:

[math]\displaystyle{ \begin{align} \mu = & \text{location parameter (also denoted as }\overline{T)} \\ \sigma = & \text{scale parameter} \end{align} }[/math]

The Logistic Mean, Median and Mode

The logistic mean or MTTF is actually one of the parameters of the distribution, usually denoted as [math]\displaystyle{ \mu }[/math] . Since the logistic distribution is symmetrical, the median and the mode are always equal to the mean, [math]\displaystyle{ \mu =\tilde{T}=\breve{T}. }[/math]

The Logistic Standard Deviation

The standard deviation of the logistic distribution, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:

[math]\displaystyle{ {{\sigma }_{T}}=\sigma \pi \frac{\sqrt{3}}{3} }[/math]

The Logistic Reliability Function

The reliability for a mission of time [math]\displaystyle{ T }[/math] , starting at age 0, for the logistic distribution is determined by:

[math]\displaystyle{ R(T)=\int_{T}^{\infty }f(t)dt }[/math]

or:

[math]\displaystyle{ R(T)=\frac{1}{1+{{e}^{z}}} }[/math]

The unreliability function is:

[math]\displaystyle{ F=\frac{{{e}^{z}}}{1+{{e}^{z}}} }[/math]

where:

[math]\displaystyle{ z=\frac{T-\mu }{\sigma } }[/math]

The Logistic Conditional Reliability Function

The logistic conditional reliability function is given by:

[math]\displaystyle{ R(t/T)=\frac{R(T+t)}{R(T)}=\frac{1+{{e}^{\tfrac{T-\mu }{\sigma }}}}{1+{{e}^{\tfrac{t+T-\mu }{\sigma }}}} }[/math]

The Logistic Reliable Life

The logistic reliable life is given by:

[math]\displaystyle{ {{T}_{R}}=\mu +\sigma [\ln (1-R)-\ln (R)] }[/math]

The Logistic Failure Rate Function

The logistic failure rate function is given by:

[math]\displaystyle{ \lambda (T)=\frac{{{e}^{z}}}{\sigma (1+{{e}^{z}})} }[/math]

Characteristics of the Logistic Distribution

• The logistic distribution has no shape parameter. This means that the logistic [math]\displaystyle{ pdf }[/math] has only one shape, the bell shape, and this shape does not change. The shape of the logistic distribution is very similar to that of the normal distribution.

• The mean, [math]\displaystyle{ \mu }[/math] , or the mean life or the [math]\displaystyle{ MTTF }[/math] , is also the location parameter of the logistic [math]\displaystyle{ pdf }[/math] , as it locates the [math]\displaystyle{ pdf }[/math] along the abscissa. It can assume values of [math]\displaystyle{ -\infty \lt \bar{T}\lt \infty }[/math] .

• As [math]\displaystyle{ \mu }[/math] decreases, the [math]\displaystyle{ pdf }[/math] is shifted to the left.

• As [math]\displaystyle{ \mu }[/math] increases, the [math]\displaystyle{ pdf }[/math] is shifted to the right.

• As [math]\displaystyle{ \sigma }[/math] decreases, the [math]\displaystyle{ pdf }[/math] gets pushed toward the mean, or it becomes narrower and taller.

• As [math]\displaystyle{ \sigma }[/math] increases, the [math]\displaystyle{ pdf }[/math] spreads out away from the mean, or it becomes broader and shallower.

• The scale parameter can assume values of [math]\displaystyle{ 0\lt \sigma \lt \infty }[/math].

• The logistic [math]\displaystyle{ pdf }[/math] starts at [math]\displaystyle{ T=-\infty }[/math] with an [math]\displaystyle{ f(T)=0 }[/math] . As [math]\displaystyle{ T }[/math] increases, [math]\displaystyle{ f(T) }[/math] also increases, goes through its point of inflection and reaches its maximum value at [math]\displaystyle{ T=\bar{T} }[/math] . Thereafter, [math]\displaystyle{ f(T) }[/math] decreases, goes through its point of inflection and assumes a value of [math]\displaystyle{ f(T)=0 }[/math] at [math]\displaystyle{ T=+\infty }[/math] .

• For [math]\displaystyle{ T=\pm \infty , }[/math] the [math]\displaystyle{ pdf }[/math] equals [math]\displaystyle{ 0. }[/math] The maximum value of the [math]\displaystyle{ pdf }[/math] occurs at [math]\displaystyle{ T }[/math] = [math]\displaystyle{ \mu }[/math] and equals [math]\displaystyle{ \tfrac{1}{4\sigma }. }[/math]

• The point of inflection of the [math]\displaystyle{ pdf }[/math] plot is the point where the second derivative of the [math]\displaystyle{ pdf }[/math] equals zero. The inflection point occurs at [math]\displaystyle{ T=\mu +\sigma \ln (2\pm \sqrt{3}) }[/math] or [math]\displaystyle{ T\approx \mu \pm \sigma 1.31696 }[/math].

• If the location parameter [math]\displaystyle{ \mu }[/math] decreases, the reliability plot is shifted to the left. If [math]\displaystyle{ \mu }[/math] increases, the reliability plot is shifted to the right.

• If [math]\displaystyle{ T=\mu }[/math] then [math]\displaystyle{ R=0.5 }[/math] . is the inflection point. If [math]\displaystyle{ T\lt \mu }[/math] then [math]\displaystyle{ R(t) }[/math] is concave (concave down); if [math]\displaystyle{ T\gt \mu }[/math] then [math]\displaystyle{ R(t) }[/math] is convex (concave up). For [math]\displaystyle{ T\lt \mu , }[/math] [math]\displaystyle{ \lambda (t) }[/math] is convex (concave up), for [math]\displaystyle{ T\gt \mu ; }[/math] [math]\displaystyle{ \lambda (t) }[/math] is concave (concave down).

• The main difference between the normal distribution and logistic distribution lies in the tails and in the behavior of the failure rate function. The logistic distribution has slightly longer tails compared to the normal distribution. Also, in the upper tail of the logistic distribution, the failure rate function levels out for large [math]\displaystyle{ t }[/math] approaching 1/ [math]\displaystyle{ \delta . }[/math]

• If location parameter [math]\displaystyle{ \mu }[/math] decreases, the failure rate plot is shifted to the left. Vice versa if [math]\displaystyle{ \mu }[/math] increases, the failure rate plot is shifted to the right.

• [math]\displaystyle{ \lambda }[/math] always increases. For [math]\displaystyle{ T\to -\infty }[/math] for [math]\displaystyle{ T\to \infty }[/math] It is always [math]\displaystyle{ 0\le \lambda (t)\le \tfrac{1}{\sigma }. }[/math]

• If [math]\displaystyle{ \sigma }[/math] increases, then [math]\displaystyle{ \lambda (t) }[/math] increases more slowly and smoothly. The segment of time where [math]\displaystyle{ 0\lt \lambda (t)\lt \tfrac{1}{\sigma } }[/math] increases, too, whereas the region where [math]\displaystyle{ \lambda (t) }[/math] is close to [math]\displaystyle{ 0 }[/math] or [math]\displaystyle{ \tfrac{1}{\sigma } }[/math] gets narrower. Conversely, if [math]\displaystyle{ \sigma }[/math] decreases, then [math]\displaystyle{ \lambda (t) }[/math] increases more quickly and sharply. The segment of time where [math]\displaystyle{ 0\lt }[/math] [math]\displaystyle{ \lambda (t)\lt \tfrac{1}{\sigma } }[/math] decreases, too, whereas the region where [math]\displaystyle{ \lambda (t) }[/math] is close to [math]\displaystyle{ 0 }[/math] or [math]\displaystyle{ \tfrac{1}{\sigma } }[/math] gets broader.

Weibull++ Notes on Negative Time Values

One of the disadvantages of using the logistic distribution for reliability calculations is the fact that the logistic distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).

Probability Paper

The form of the Logistic probability paper is based on linearizing the [math]\displaystyle{ cdf }[/math] . From Eqn. (UnR fcn), [math]\displaystyle{ z }[/math] can be calculated as a function of the [math]\displaystyle{ cdf }[/math] [math]\displaystyle{ F }[/math] as follows:

[math]\displaystyle{ z=\ln (F)-\ln (1-F) }[/math]

or using Eqn. (z func of parameters)

[math]\displaystyle{ \frac{T-\mu }{\sigma }=\ln (F)-\ln (1-F) }[/math]

Then:

[math]\displaystyle{ \ln (F)-\ln (1-F)=-\frac{\mu }{\sigma }+\frac{1}{\sigma }T }[/math]

Now let:

[math]\displaystyle{ y=\ln (F)-\ln (1-F) }[/math]

[math]\displaystyle{ x=T }[/math]

and:

[math]\displaystyle{ a=-\frac{\mu }{\sigma } }[/math]

[math]\displaystyle{ b=\frac{1}{\sigma } }[/math]

which results in the following linear equation:

[math]\displaystyle{ y=a+bx }[/math]

The logistic probability paper resulting from this linearized [math]\displaystyle{ cdf }[/math] function is shown next.

Since the logistic distribution is symmetrical, the area under the [math]\displaystyle{ pdf }[/math] curve from [math]\displaystyle{ -\infty }[/math] to [math]\displaystyle{ \mu }[/math] is [math]\displaystyle{ 0.5 }[/math] , as is the area from [math]\displaystyle{ \mu }[/math] to [math]\displaystyle{ +\infty }[/math] . Consequently, the value of [math]\displaystyle{ \mu }[/math] is said to be the point where [math]\displaystyle{ R(t)=Q(t)=50% }[/math] . This means that the estimate of [math]\displaystyle{ \mu }[/math] can be read from the point where the plotted line crosses the 50% unreliability line. For [math]\displaystyle{ z=1 }[/math] , [math]\displaystyle{ \sigma =t-\mu }[/math] and [math]\displaystyle{ R(t)=\tfrac{1}{1+\exp (1)}\approx 0.2689. }[/math] Therefore, [math]\displaystyle{ \sigma }[/math] can be found by subtracting [math]\displaystyle{ \mu }[/math] from the time value where the plotted probability line crosses the 73.10% unreliability (26.89% reliability) horizontal line.

Confidence Bounds

In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in Chapter 5.

Bounds on the Parameters

The lower and upper bounds on the location parameter [math]\displaystyle{ \widehat{\mu } }[/math] are estimated from

[math]\displaystyle{ {{\mu }_{U}}=\widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (upper bound)} }[/math]

[math]\displaystyle{ {{\mu }_{L}}=\widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })\text{ }}\text{ (lower bound)} }[/math]

The lower and upper bounds on the scale parameter [math]\displaystyle{ \widehat{\sigma } }[/math] are estimated from:

[math]\displaystyle{ {{\sigma }_{U}}=\widehat{\sigma }{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}(\text{upper bound}) }[/math]

[math]\displaystyle{ {{\sigma }_{L}}=\widehat{\sigma }{{e}^{\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })\text{ }}}{\widehat{\sigma }}}}\text{ (lower bound)} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variances and covariances of [math]\displaystyle{ \widehat{\mu } }[/math] and [math]\displaystyle{ \widehat{\sigma } }[/math] are estimated from the Fisher matrix, as follows:

[math]\displaystyle{ \left( \begin{matrix} \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) \\ \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\sigma } \right) \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } \\ {} & {} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} \\ \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1} }[/math]

[math]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the normal distribution, described in Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the logistic distribution is:

[math]\displaystyle{ \widehat{R}=\frac{1}{1+{{e}^{\widehat{z}}}} }[/math]

where:

[math]\displaystyle{ \widehat{z}=\frac{T-\widehat{\mu }}{\widehat{\sigma }} }[/math]

Here [math]\displaystyle{ -\infty \lt T\lt \infty }[/math] , [math]\displaystyle{ -\infty \lt \mu \lt \infty }[/math] , [math]\displaystyle{ 0\lt \sigma \lt \infty }[/math] . Therefore, [math]\displaystyle{ z }[/math] also is changing from [math]\displaystyle{ -\infty }[/math] to [math]\displaystyle{ +\infty }[/math] . Then the bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ {{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }} }[/math]

[math]\displaystyle{ {{z}_{L}}=\widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}\text{ } }[/math]

where:

[math]\displaystyle{ Var(\widehat{z})={{(\frac{\partial z}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial z}{\partial \mu })(\frac{\partial z}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial z}{\partial \sigma })}^{2}}Var(\widehat{\sigma }) }[/math]

or:

[math]\displaystyle{ Var(\widehat{z})=\frac{1}{{{\sigma }^{2}}}(Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })) }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ {{R}_{U}}=\frac{1}{1+{{e}^{{{z}_{L}}}}}\text{(upper bound)} }[/math]

[math]\displaystyle{ {{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(lower bound)} }[/math]

Bounds on Time

The bounds around time for a given logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:

[math]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]

where:

[math]\displaystyle{ z=\ln (1-R)-\ln (R) }[/math]

[math]\displaystyle{ Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma }) }[/math]

or:

[math]\displaystyle{ Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma }) }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ {{T}_{U}}=\widehat{T}+{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{upper bound}) }[/math]

[math]\displaystyle{ {{T}_{L}}=\widehat{T}-{{K}_{\alpha }}\sqrt{Var(\widehat{T})\text{ }}(\text{lower bound}) }[/math]

A Logistic Distribution Example

The lifetime of a mechanical valve is known to follow a logistic distribution. Ten units were tested for 28 months and the following months-to-failure data was collected.

[math]\displaystyle{ \overset{{}}{\mathop{\text{Table 10}\text{.1 - Times-to-Failure Data with Suspensions}}}\, }[/math]

[math]\displaystyle{ \begin{matrix} \text{Data Point Index} & \text{State F or S} & \text{State End Time} \\ \text{1} & \text{F} & \text{8} \\ \text{2} & \text{F} & \text{10} \\ \text{3} & \text{F} & \text{15} \\ \text{4} & \text{F} & \text{17} \\ \text{5} & \text{F} & \text{19} \\ \text{6} & \text{F} & \text{26} \\ \text{7} & \text{F} & \text{27} \\ \text{8} & \text{S} & \text{28} \\ \text{9} & \text{S} & \text{28} \\ \text{10} & \text{S} & \text{28} \\ \end{matrix} }[/math]

• Determine the valve's design life if specifications call for a reliability goal of 0.90.

• The valve is to be used in a pumping device that requires 1 month of continuous operation. What is the probability of the pump failing due to the valve?

This data set can be entered into Weibull++ as follows:

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 22.34 \\ & \hat{\sigma }= & 6.15 \end{align} }[/math]

• The valve's design life, along with 90% two sided confidence bounds, can be obtained using the QCP as follows:

• The probability, along with 90% two sided confidence bounds, that the pump fails due to a valve failure during the first month is obtained as follows:

The Loglogistic Distribution

As may be indicated by the name, the loglogistic distribution has certain similarities to the logistic distribution. A random variable is loglogistically distributed if the logarithm of the random variable is logistically distributed. Because of this, there are many mathematical similarities between the two distributions [27]. For example, the mathematical reasoning for the construction of the probability plotting scales is very similar for these two distributions.

Loglogistic Probability Density Function

The loglogistic distribution is a two-parameter distribution with parameters [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \sigma }[/math] . The [math]\displaystyle{ pdf }[/math] for this distribution is given by:

[math]\displaystyle{ f(T)=\frac{{{e}^{z}}}{\sigma T{{(1+{{e}^{z}})}^{2}}} }[/math]

where:

[math]\displaystyle{ z=\frac{{T}'-\mu }{\sigma } }[/math]

[math]\displaystyle{ {T}'=\ln (T) }[/math]

and:

[math]\displaystyle{ \begin{align} & \mu = & \text{scale parameter} \\ & \sigma = & \text{shape parameter} \end{align} }[/math]

where [math]\displaystyle{ 0\lt t\lt \infty }[/math] , [math]\displaystyle{ -\infty \lt \mu \lt \infty }[/math] and [math]\displaystyle{ 0\lt \sigma \lt \infty }[/math] .

Mean, Median and Mode

The mean of the loglogistic distribution, [math]\displaystyle{ \overline{T} }[/math] , is given by:

[math]\displaystyle{ \overline{T}={{e}^{\mu }}\Gamma (1+\sigma )\Gamma (1-\sigma ) }[/math]

Note that for [math]\displaystyle{ \sigma \ge 1, }[/math] [math]\displaystyle{ \overline{T} }[/math] does not exist.

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

[math]\displaystyle{ \widehat{T}={{e}^{\mu }} }[/math]

The mode of the loglogistic distribution, [math]\displaystyle{ \tilde{T} }[/math] , if [math]\displaystyle{ \sigma \lt 1, }[/math] is given by:

..

The Standard Deviation

The standard deviation of the loglogistic distribution, [math]\displaystyle{ {{\sigma }_{T}} }[/math] , is given by:

[math]\displaystyle{ {{\sigma }_{T}}={{e}^{\mu }}\sqrt{\Gamma (1+2\sigma )\Gamma (1-2\sigma )-{{(\Gamma (1+\sigma )\Gamma (1-\sigma ))}^{2}}} }[/math]

Note that for [math]\displaystyle{ \sigma \ge 0.5, }[/math] the standard deviation does not exist.

The Loglogistic Reliability Function

The reliability for a mission of time [math]\displaystyle{ T }[/math] , starting at age 0, for the loglogistic distribution is determined by:

[math]\displaystyle{ R=\frac{1}{1+{{e}^{z}}} }[/math]

where:

[math]\displaystyle{ z=\frac{{T}'-\mu }{\sigma } }[/math]

[math]\displaystyle{ {T}'=\ln (t) }[/math]

The unreliability function is:

[math]\displaystyle{ F=\frac{{{e}^{z}}}{1+{{e}^{z}}} }[/math]

The loglogistic Reliable Life

The logistic reliable life is:

[math]\displaystyle{ {{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}} }[/math]

The loglogistic Failure Rate Function

The loglogistic failure rate is given by:

[math]\displaystyle{ \lambda (T)=\frac{{{e}^{z}}}{\sigma T(1+{{e}^{z}})} }[/math]

Distribution Characteristics

For [math]\displaystyle{ \sigma \gt 1 }[/math] :

• [math]\displaystyle{ f(T) }[/math] decreases monotonically and is convex. Mode and mean do not exist.

For [math]\displaystyle{ \sigma =1 }[/math] :

• [math]\displaystyle{ f(T) }[/math] decreases monotonically and is convex. Mode and mean do not exist. As [math]\displaystyle{ T\to 0 }[/math] , [math]\displaystyle{ f(T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}. }[/math]

• As [math]\displaystyle{ T\to 0 }[/math] , [math]\displaystyle{ \lambda (T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}. }[/math]

For [math]\displaystyle{ 0\lt \sigma \lt 1 }[/math] :

• The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.

• The [math]\displaystyle{ pdf }[/math] starts at zero, increases to its mode, and decreases thereafter.

• As [math]\displaystyle{ \mu }[/math] increases, while [math]\displaystyle{ \sigma }[/math] is kept the same, the [math]\displaystyle{ pdf }[/math] gets stretched out to the right and its height decreases, while maintaining its shape.

• As [math]\displaystyle{ \mu }[/math] decreases,while [math]\displaystyle{ \sigma }[/math] is kept the same, the .. gets pushed in towards the left and its height increases.

• [math]\displaystyle{ \lambda (T) }[/math] increases till [math]\displaystyle{ T={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}} }[/math] and decreases thereafter. [math]\displaystyle{ \lambda (T) }[/math] is concave at first, then becomes convex.

Confidence Bounds

The method used by the application in estimating the different types of confidence bounds for loglogistically distributed data is presented in this section. The complete derivations were presented in detail for a general function in Chapter 5.

Bounds on the Parameters

The lower and upper bounds on the mean, [math]\displaystyle{ {\mu }' }[/math] , are estimated from:

[math]\displaystyle{ \begin{align} & \mu _{U}^{\prime }= & {{\widehat{\mu }}^{\prime }}+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\ & \mu _{L}^{\prime }= & {{\widehat{\mu }}^{\prime }}-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)} \end{align} }[/math]

For the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] is treated as normally distributed, and the bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}\text{ (upper bound)} \\ & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (lower bound)} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds.

The variances and covariances of [math]\displaystyle{ \widehat{\mu } }[/math] and [math]\displaystyle{ \widehat{\sigma } }[/math] are estimated as follows:

[math]\displaystyle{ \left( \begin{matrix} \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) \\ \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\sigma } \right) \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{(\mu )}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } \\ {} & {} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} \\ \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1} }[/math]

where [math]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the loglogistic distribution.

Bounds on Reliability

The reliability of the logistic distribution is:

[math]\displaystyle{ \widehat{R}=\frac{1}{1+\exp (\widehat{z})} }[/math]

where:

[math]\displaystyle{ \widehat{z}=\frac{{T}'-\widehat{\mu }}{\widehat{\sigma }} }[/math]

Here [math]\displaystyle{ 0\lt t\lt \infty }[/math] , [math]\displaystyle{ -\infty \lt \mu \lt \infty }[/math] , [math]\displaystyle{ 0\lt \sigma \lt \infty }[/math] , therefore [math]\displaystyle{ 0\lt \ln (t)\lt \infty }[/math] and [math]\displaystyle{ z }[/math] also is changing from [math]\displaystyle{ -\infty }[/math] till [math]\displaystyle{ +\infty }[/math] .The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ {{z}_{U}}=\widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} }[/math]

[math]\displaystyle{ {{z}_{L}}=\widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})\text{ }}\text{ } }[/math]

where:

[math]\displaystyle{ Var(\widehat{z})={{(\frac{\partial z}{\partial \mu })}^{2}}Var({{\widehat{\mu }}^{\prime }})+2(\frac{\partial z}{\partial \mu })(\frac{\partial z}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial z}{\partial \sigma })}^{2}}Var(\widehat{\sigma }) }[/math]

or:

[math]\displaystyle{ Var(\widehat{z})=\frac{1}{{{\sigma }^{2}}}(Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })) }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ {{R}_{U}}=\frac{1}{1+{{e}^{{{z}_{L}}}}}\text{(Upper bound)} }[/math]

[math]\displaystyle{ {{R}_{L}}=\frac{1}{1+{{e}^{{{z}_{U}}}}}\text{(Lower bound)} }[/math]

Bounds on Time

The bounds around time for a given loglogistic percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })={{e}^{\widehat{\mu }+\widehat{\sigma }z}} }[/math]

where:

[math]\displaystyle{ z=\ln (1-R)-\ln (R) }[/math]

or:

[math]\displaystyle{ \ln (T)=\widehat{\mu }+\widehat{\sigma }z }[/math]

Let:

[math]\displaystyle{ u=\ln (T)=\widehat{\mu }+\widehat{\sigma }z }[/math]

then:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ } }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})\text{ }}\text{ } }[/math]

where:

[math]\displaystyle{ Var(\widehat{u})={{(\frac{\partial u}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial u}{\partial \mu })(\frac{\partial u}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial u}{\partial \sigma })}^{2}}Var(\widehat{\sigma }) }[/math]

or:

[math]\displaystyle{ Var(\widehat{u})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma }) }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ {{T}_{U}}={{e}^{{{u}_{U}}}}\text{ (upper bound)} }[/math]

[math]\displaystyle{ {{T}_{L}}={{e}^{{{u}_{L}}}}\text{ (lower bound)} }[/math]

A LogLogistic Distribution Example

Determine the loglogistic parameter estimates for the data given in Table 10.3.

[math]\displaystyle{ \overset{{}}{\mathop{\text{Table 10}\text{.3 - Test data}}}\, }[/math]

[math]\displaystyle{ \begin{matrix} \text{Data point index} & \text{Last Inspected} & \text{State End time} \\ \text{1} & \text{105} & \text{106} \\ \text{2} & \text{197} & \text{200} \\ \text{3} & \text{297} & \text{301} \\ \text{4} & \text{330} & \text{335} \\ \text{5} & \text{393} & \text{401} \\ \text{6} & \text{423} & \text{426} \\ \text{7} & \text{460} & \text{468} \\ \text{8} & \text{569} & \text{570} \\ \text{9} & \text{675} & \text{680} \\ \text{10} & \text{884} & \text{889} \\ \end{matrix} }[/math]

Using Times-to-failure data under the Folio Data Type and the My data set contains interval and/or left censored data under Times-to-failure data options to enter the above data, the computed parameters for maximum likelihood are calculated to be:

[math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 5.9772 \\ & {{{\hat{\sigma }}}_{{{T}'}}}= & 0.3256 \end{align} }[/math]

For rank regression on [math]\displaystyle{ X\ \ : }[/math]

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 5.9281 \\ & \hat{\sigma }= & 0.3821 \end{align} }[/math]

For rank regression on [math]\displaystyle{ Y\ \ : }[/math]

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 5.9772 \\ & \hat{\sigma }= & 0.3256 \end{align} }[/math]

The Gumbel/SEV Distribution

The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type I) distribution. The Gumbel distribution's [math]\displaystyle{ pdf }[/math] is skewed to the left, unlike the Weibull distribution's [math]\displaystyle{ pdf }[/math] , which is skewed to the right. The Gumbel distribution is appropriate for modeling strength, which is sometimes skewed to the left (few weak units in the lower tail, most units in the upper tail of the strength population). The Gumbel distribution could also be appropriate for modeling the life of products that experience very quick wear-out after reaching a certain age. The distribution of logarithms of times can often be modeled with the Gumbel distribution (in addition to the more common lognormal distribution). [27]

Gumbel Probability Density Function

The [math]\displaystyle{ pdf }[/math] of the Gumbel distribution is given by:

[math]\displaystyle{ f(T)=\frac{1}{\sigma }{{e}^{z-{{e}^{z}}}} }[/math]

[math]\displaystyle{ f(T)\ge 0,\sigma \gt 0 }[/math] where:

[math]\displaystyle{ z=\frac{T-\mu }{\sigma } }[/math]

and:

[math]\displaystyle{ \begin{align} & \mu = & \text{location parameter} \\ & \sigma = & \text{scale parameter} \end{align} }[/math]

The Gumbel Mean, Median and Mode

The Gumbel mean or MTTF is:

[math]\displaystyle{ \overline{T}=\mu -\sigma \gamma }[/math]

where [math]\displaystyle{ \gamma \approx 0.5772 }[/math] (Euler's constant).

The mode of the Gumbel distribution is:

[math]\displaystyle{ \tilde{T}=\mu }[/math]

The median of the Gumbel distribution is:

[math]\displaystyle{ \widehat{T}=\mu +\sigma \ln (\ln (2)) }[/math]

The Gumbel Standard Deviation

The standard deviation for the Gumbel distribution is given by:

[math]\displaystyle{ {{\sigma }_{T}}=\sigma \pi \frac{\sqrt{6}}{6} }[/math]

The Gumbel Reliability Function

The reliability for a mission of time [math]\displaystyle{ T }[/math] for the Gumbel distribution is given by:

[math]\displaystyle{ R(T)={{e}^{-{{e}^{z}}}} }[/math]

The unreliability function is given by:

[math]\displaystyle{ F(T)=1-{{e}^{-{{e}^{z}}}} }[/math]

The Gumbel Reliable Life

The Gumbel reliable life is given by:

[math]\displaystyle{ {{T}_{R}}=\mu +\sigma [\ln (-\ln (R))] }[/math]

The Gumbel Failure Rate Function

The instantaneous Gumbel failure rate is given by:

[math]\displaystyle{ \lambda =\frac{{{e}^{z}}}{\sigma } }[/math]

Characteristics of the Gumbel Distribution

Some of the specific characteristics of the Gumbel distribution are the following:

• The shape of the Gumbel distribution is skewed to the left. The Gumbel [math]\displaystyle{ pdf }[/math] has no shape parameter. This means that the Gumbel [math]\displaystyle{ pdf }[/math] has only one shape, which does not change.

• The Gumbel [math]\displaystyle{ pdf }[/math] has location parameter [math]\displaystyle{ \mu , }[/math] which is equal to the mode [math]\displaystyle{ \tilde{T}, }[/math] but it differs from median and mean. This is because the Gumbel distribution is not symmetrical about its [math]\displaystyle{ \mu }[/math] .

• As [math]\displaystyle{ \mu }[/math] decreases, the [math]\displaystyle{ pdf }[/math] is shifted to the left.

• As [math]\displaystyle{ \mu }[/math] increases, the [math]\displaystyle{ pdf }[/math] is shifted to the right.

• As [math]\displaystyle{ \sigma }[/math] increases, the [math]\displaystyle{ pdf }[/math] spreads out and becomes shallower.

• As [math]\displaystyle{ \sigma }[/math] decreases, the [math]\displaystyle{ pdf }[/math] becomes taller and narrower.

• For [math]\displaystyle{ T=\pm \infty , }[/math] [math]\displaystyle{ pdf=0. }[/math] For [math]\displaystyle{ T=\mu }[/math] , the [math]\displaystyle{ pdf }[/math] reaches its maximum point [math]\displaystyle{ \frac{1}{\sigma e} }[/math]

• The points of inflection of the [math]\displaystyle{ pdf }[/math] graph are [math]\displaystyle{ T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2}) }[/math] or [math]\displaystyle{ T\approx \mu \pm \sigma 0.96242 }[/math] .

• If times follow the Weibull distribution, then the logarithm of times follow a Gumbel distribution. If [math]\displaystyle{ {{t}_{i}} }[/math] follows a Weibull distribution with [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \eta }[/math] , then the [math]\displaystyle{ Ln({{t}_{i}}) }[/math] follows a Gumbel distribution with [math]\displaystyle{ \mu =\ln (\eta ) }[/math] and [math]\displaystyle{ \sigma =\tfrac{1}{\beta } }[/math] [32] [math]\displaystyle{ . }[/math]

Probability Paper

The form of the Gumbel probability paper is based on a linearization of the [math]\displaystyle{ cdf }[/math] . From Eqn. (UnrGumbel):

[math]\displaystyle{ z=\ln (-\ln (1-F)) }[/math]

using Eqns. (z3):

[math]\displaystyle{ \frac{T-\mu }{\sigma }=\ln (-\ln (1-F)) }[/math]

Then:

[math]\displaystyle{ \ln (-\ln (1-F))=-\frac{\mu }{\sigma }+\frac{1}{\sigma }T }[/math]

Now let:

[math]\displaystyle{ y=\ln (-\ln (1-F)) }[/math]

[math]\displaystyle{ x=T }[/math]

and:

[math]\displaystyle{ \begin{align} & a= & -\frac{\mu }{\sigma } \\ & b= & \frac{1}{\sigma } \end{align} }[/math]

which results in the linear equation of:

[math]\displaystyle{ y=a+bx }[/math]

The Gumbel probability paper resulting from this linearized [math]\displaystyle{ cdf }[/math] function is shown next.

For [math]\displaystyle{ z=0 }[/math] , [math]\displaystyle{ T=\mu }[/math] and [math]\displaystyle{ R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678 }[/math] (63.21% unreliability). For [math]\displaystyle{ z=1 }[/math] , [math]\displaystyle{ \sigma =T-\mu }[/math] and [math]\displaystyle{ R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659. }[/math] To read [math]\displaystyle{ \mu }[/math] from the plot, find the time value that corresponds to the intersection of the probability plot with the 63.21% unreliability line. To read [math]\displaystyle{ \sigma }[/math] from the plot, find the time value that corresponds to the intersection of the probability plot with the 93.40% unreliability line, then take the difference between this time value and the [math]\displaystyle{ \mu }[/math] value.

Confidence Bounds

This section presents the method used by the application to estimate the different types of confidence bounds for data that follow the Gumbel distribution. The complete derivations were presented in detail (for a general function) in Chapter 5. Only Fisher Matrix confidence bounds are available for the Gumbel distribution.

Bounds on the Parameters

The lower and upper bounds on the mean, [math]\displaystyle{ \widehat{\mu } }[/math] , are estimated from:

[math]\displaystyle{ \begin{align} & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\ & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)} \end{align} }[/math]

Since the standard deviation, [math]\displaystyle{ \widehat{\sigma } }[/math] , must be positive, then [math]\displaystyle{ \ln (\widehat{\sigma }) }[/math] is treated as normally distributed, and the bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{\sigma }_{U}}= & \widehat{\sigma }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{{{\widehat{\sigma }}_{T}}}}}\text{ (upper bound)} \\ & {{\sigma }_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}}\text{ (lower bound)} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds.

The variances and covariances of [math]\displaystyle{ \widehat{\mu } }[/math] and [math]\displaystyle{ \widehat{\sigma } }[/math] are estimated from the Fisher matrix as follows:

[math]\displaystyle{ \left( \begin{matrix} \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) \\ \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\sigma } \right) \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } \\ {} & {} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} \\ \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1} }[/math]

[math]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the Gumbel distribution, described in Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the Gumbel distribution is given by:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{\sigma })={{e}^{-{{e}^{{\hat{z}}}}}} }[/math]

where:

[math]\displaystyle{ \widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }} }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ Var(\widehat{z})={{\left( \frac{\partial z}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial z}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+2\left( \frac{\partial z}{\partial \mu } \right)\left( \frac{\partial z}{\partial \sigma } \right)Cov\left( \widehat{\mu },\widehat{\sigma } \right) }[/math]

or:

[math]\displaystyle{ Var(\widehat{z})=\frac{1}{{{\widehat{\sigma }}^{2}}}\left[ Var(\widehat{\mu })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })+2\cdot \widehat{z}\cdot Cov\left( \widehat{\mu },\widehat{\sigma } \right) \right] }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & {{e}^{-{{e}^{{{z}_{L}}}}}}\text{ (upper bound)} \\ & {{R}_{L}}= & {{e}^{-{{e}^{{{z}_{U}}}}}}\text{ (lower bound)} \end{align} }[/math]

Bounds on Time

The bounds around time for a given Gumbel percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z }[/math]

where:

[math]\displaystyle{ z=\ln (-\ln (R)) }[/math]

[math]\displaystyle{ Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma }) }[/math]

or:

[math]\displaystyle{ Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma }) }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\ & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)} \end{align} }[/math]

A Gumbel Distribution Example

Verify using Monte Carlo simulation that if [math]\displaystyle{ {{t}_{i}} }[/math] follows a Weibull distribution with [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \eta }[/math] , then the [math]\displaystyle{ Ln({{t}_{i}}) }[/math] follows a Gumbel distribution with [math]\displaystyle{ \mu =\ln (\eta ) }[/math] and [math]\displaystyle{ \sigma =1/\beta ). }[/math] Let us assume that [math]\displaystyle{ {{t}_{i}} }[/math] follows a Weibull distribution with [math]\displaystyle{ \beta =0.5 }[/math] and [math]\displaystyle{ \eta =10000. }[/math] The Monte Carlo simulation tool in Weibull++ can be used to generate a set of random numbers that follow a Weibull distribution with the specified parameters.

After obtaining the random time values [math]\displaystyle{ {{t}_{i}} }[/math] , insert a new Data Sheet using the Insert Data Sheet option under the Folio menu. In this sheet enter the [math]\displaystyle{ Ln({{t}_{i}}) }[/math] values using the LN function and referring to the cells in the sheet that contains the [math]\displaystyle{ {{t}_{i}} }[/math] values. Delete any negative values, if there are any, since Weibull++ expects time values to be positive. Calculate the parameters of the Gumbel distribution that fits the [math]\displaystyle{ Ln({{t}_{i}}) }[/math] values.

Using maximum likelihood as the analysis method, the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= & 9.3816 \\ & \hat{\sigma }= & 1.9717 \end{align} }[/math]

Since [math]\displaystyle{ \ln (\eta )= }[/math] 9.2103 ( [math]\displaystyle{ \simeq 9.3816 }[/math] ) and [math]\displaystyle{ 1/\beta =2 }[/math] [math]\displaystyle{ (\simeq 1.9717), }[/math] then this simulation verifies that [math]\displaystyle{ Ln({{t}_{i}}) }[/math] follows a Gumbel distribution with [math]\displaystyle{ \mu =\ln (\eta ) }[/math] and [math]\displaystyle{ \delta =1/\beta . }[/math] Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.