Template:Lognormal distribution estimation of the parameters
Estimation of the Parameters
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, as discussed in Meeker and Escobar [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 2-parameter distribution with parameters [math]\displaystyle{ \mu \,\! }[/math] and [math]\displaystyle{ \sigma \,\! }[/math]. The pdf 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:
- [math]\displaystyle{ \tilde{T} = e^{\mu+\sigma ln(\frac{1-\sigma}{1+\sigma})}\,\! }[/math]
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{ \begin{align} {t}'=\ln (t) \end{align}\,\! }[/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{ \begin{align} {{T}_{R}}={{e}^{\mu +\sigma [\ln (1-R)-\ln (R)]}} \end{align}\,\! }[/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 pdf 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 pdf 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 pdf 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 Parameter Estimation.
Bounds on the Parameters
The lower and upper bounds [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 paramter [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]
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 t'=\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{ \begin{align} z=\ln (1-R)-\ln (R) \end{align}\,\! }[/math]
or:
- [math]\displaystyle{ \ln (\hat{T})=\widehat{\mu }+\widehat{\sigma }z\,\! }[/math]
Let:
- [math]\displaystyle{ {u}=\ln (\hat{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]
General Examples
Determine the loglogistic parameter estimates for the data given in the following table.
Set up the folio for times-to-failure data that includes interval and left censored data, then enter the 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 lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units.
As may be surmised by the name, the lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Because of this, there are many mathematical similarities between the two distributions. For example, the mathematical reasoning for the construction of the probability plotting scales and the bias of parameter estimators is very similar for these two distributions.
Lognormal Probability Density Function
The lognormal distribution is a 2-parameter distribution with parameters [math]\displaystyle{ {\mu }'\,\! }[/math] and [math]\displaystyle{ \sigma'\,\! }[/math]. The pdf for this distribution is given by:
- [math]\displaystyle{ f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}\,\! }[/math]
where:
- [math]\displaystyle{ {t}'=\ln (t)\,\! }[/math]. [math]\displaystyle{ t\,\! }[/math] values are the times-to-failure
- [math]\displaystyle{ \mu'\,\! }[/math] = mean of the natural logarithms of the times-to-failure
- [math]\displaystyle{ \sigma'\,\! }[/math] = standard deviation of the natural logarithms of the times-to-failure
The lognormal pdf can be obtained, realizing that for equal probabilities under the normal and lognormal pdfs, incremental areas should also be equal, or:
- [math]\displaystyle{ \begin{align} f(t)dt=f({t}')d{t}' \end{align}\,\! }[/math]
Taking the derivative of the relationship between [math]\displaystyle{ {t}'\,\! }[/math] and [math]\displaystyle{ {t}\,\! }[/math] yields:
- [math]\displaystyle{ d{t}'=\frac{dt}{t}\,\! }[/math]
Substitution yields:
- [math]\displaystyle{ \begin{align} f(t)= & \frac{f({t}')}{t} \\ f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} \end{align}\,\! }[/math]
where:
- [math]\displaystyle{ f(t)\ge 0,t\gt 0,-\infty \lt {\mu }'\lt \infty ,{{\sigma' }}\gt 0\,\! }[/math]
Lognormal Distribution Functions
The Mean or MTTF
The mean of the lognormal distribution, [math]\displaystyle{ \mu \,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}\,\! }[/math]
The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ \mu'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {{\sigma}}\,\! }[/math] is given by:
- [math]\displaystyle{ {\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma^{2}}{{{{\bar{T}}}^{2}}}+1 \right)\,\! }[/math]
The Median
The median of the lognormal distribution, [math]\displaystyle{ \breve{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \breve{T}={{e}^{{{\mu}'}}}\,\! }[/math]
The Mode
The mode of the lognormal distribution, [math]\displaystyle{ \tilde{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \tilde{T}={{e}^{{\mu }'-\sigma'^{2}}}\,\! }[/math]
The Standard Deviation
The standard deviation of the lognormal distribution, [math]\displaystyle{ {\sigma }_{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ {\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}\,\! }[/math]
The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {\sigma}'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {\sigma}\,\! }[/math] is given by:
- [math]\displaystyle{ \sigma '=\sqrt{\ln \left( \frac{{\sigma}_{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}\,\! }[/math]
The Lognormal Reliability Function
The reliability for a mission of time [math]\displaystyle{ t\,\! }[/math], starting at age 0, for the lognormal distribution is determined by:
- [math]\displaystyle{ R(t)=\int_{t}^{\infty }f(x)dx\,\! }[/math]
or:
- [math]\displaystyle{ {{R}({t})}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
As with the normal distribution, there is no closed-form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.
The Lognormal Conditional Reliability Function
The lognormal conditional reliability function is given by:
- [math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{\text{ln}(T+t)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx}\,\! }[/math]
Once again, the use of standard normal tables is necessary to solve this equation, as no closed-form solution exists.
The Lognormal Reliable Life Function
As there is no closed-form solution for the lognormal reliability equation, no closed-form solution exists for the lognormal reliable life either. In order to determine this value, one must solve the following equation for [math]\displaystyle{ t\,\! }[/math]:
- [math]\displaystyle{ {{R}_{t}}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
The Lognormal Failure Rate Function
The lognormal failure rate is given by:
- [math]\displaystyle{ \lambda (t)=\frac{f(t)}{R(t)}=\frac{\tfrac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{t}'-{\mu }'}{{{\sigma' }}})}^{2}}}}}{\int_{{{t}'}}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{x-{\mu }'}{{{\sigma' }}})}^{2}}}}dx}\,\! }[/math]
As with the reliability equations, standard normal tables will be required to solve for this function.
Characteristics of the Lognormal Distribution
- The lognormal distribution is a distribution skewed to the right.
- The pdf starts at zero, increases to its mode, and decreases thereafter.
- The degree of skewness increases as [math]\displaystyle{ {{\sigma'}}\,\! }[/math] increases, for a given [math]\displaystyle{ \mu'\,\! }[/math]
- For the same [math]\displaystyle{ {{\sigma'}}\,\! }[/math], the pdf 's skewness increases as [math]\displaystyle{ {\mu }'\,\! }[/math] increases.
- For [math]\displaystyle{ {{\sigma' }}\,\! }[/math] values significantly greater than 1, the pdf rises very sharply in the beginning, (i.e., for very small values of [math]\displaystyle{ T\,\! }[/math] near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential pdf or a Weibull pdf with [math]\displaystyle{ 0\lt \beta \lt 1\,\! }[/math].
- The parameter, [math]\displaystyle{ {\mu }'\,\! }[/math], in terms of the logarithm of the [math]\displaystyle{ {T}'s\,\! }[/math] is also the scale parameter, and not the location parameter as in the case of the normal pdf.
- The parameter [math]\displaystyle{ {{\sigma'}}\,\! }[/math], or the standard deviation of the [math]\displaystyle{ {T}'s\,\! }[/math] in terms of their logarithm or of their [math]\displaystyle{ {T}'\,\! }[/math], is also the shape parameter and not the scale parameter, as in the normal pdf, and assumes only positive values.
Lognormal Distribution Parameters in ReliaSoft's Software
In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter [math]\displaystyle{ {\mu }'\,\! }[/math] represents the mean of the natural logarithms of the times-to-failure, while [math]\displaystyle{ {{\sigma' }}\,\! }[/math] represents the standard deviation of these data point logarithms. Specifically, the returned [math]\displaystyle{ {{\sigma' }}\,\! }[/math] is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.
Lognormal Distribution Examples
Complete Data Example
Determine the lognormal parameter estimates for the data given in the following table.
| Non-Grouped Times-to-Failure Data | ||
| Data point index | State F or S | State End Time |
|---|---|---|
| 1 | F | 2 |
| 2 | F | 5 |
| 3 | F | 11 |
| 4 | F | 23 |
| 5 | F | 29 |
| 6 | F | 37 |
| 7 | F | 43 |
| 8 | F | 59 |
Solution
Using Weibull++, the computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {\hat{\sigma '}}= & 1.10 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.24 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y:\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.36 \end{align}\,\! }[/math]
Complete Data RRX Example
From Kececioglu [20, p. 347]. 15 identical units were tested to failure and following is a table of their failure times:
Solution
Published results (using probability plotting):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.22575 \\ {{\widehat{\sigma' }}}=0.62048. \\ \end{matrix}\,\! }[/math]
Weibull++ computed parameters for rank regression on X are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.2303 \\ {{\widehat{\sigma'}}}=0.6283. \\ \end{matrix}\,\! }[/math]
The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.
Complete Data Unbiased MLE Example
From Kececioglu [19, p. 406]. 9 identical units are tested continuously to failure and failure times were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1 and 257.9 hours.
Solution
The results published were obtained by using the unbiased model. Published Results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.67677 \\ \end{matrix}\,\! }[/math]
This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results.
Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.6768 \\ \end{matrix}\,\! }[/math]
Suspension Data Example
From Nelson [30, p. 324]. 96 locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. The table below shows the failure and suspension times.
| Nelson's Locomotive Data | |||
| Number in State | F or S | Time | |
|---|---|---|---|
| 1 | 1 | F | 22.5 |
| 2 | 1 | F | 37.5 |
| 3 | 1 | F | 46 |
| 4 | 1 | F | 48.5 |
| 5 | 1 | F | 51.5 |
| 6 | 1 | F | 53 |
| 7 | 1 | F | 54.5 |
| 8 | 1 | F | 57.5 |
| 9 | 1 | F | 66.5 |
| 10 | 1 | F | 68 |
| 11 | 1 | F | 69.5 |
| 12 | 1 | F | 76.5 |
| 13 | 1 | F | 77 |
| 14 | 1 | F | 78.5 |
| 15 | 1 | F | 80 |
| 16 | 1 | F | 81.5 |
| 17 | 1 | F | 82 |
| 18 | 1 | F | 83 |
| 19 | 1 | F | 84 |
| 20 | 1 | F | 91.5 |
| 21 | 1 | F | 93.5 |
| 22 | 1 | F | 102.5 |
| 23 | 1 | F | 107 |
| 24 | 1 | F | 108.5 |
| 25 | 1 | F | 112.5 |
| 26 | 1 | F | 113.5 |
| 27 | 1 | F | 116 |
| 28 | 1 | F | 117 |
| 29 | 1 | F | 118.5 |
| 30 | 1 | F | 119 |
| 31 | 1 | F | 120 |
| 32 | 1 | F | 122.5 |
| 33 | 1 | F | 123 |
| 34 | 1 | F | 127.5 |
| 35 | 1 | F | 131 |
| 36 | 1 | F | 132.5 |
| 37 | 1 | F | 134 |
| 38 | 59 | S | 135 |
Solution
The distribution used in the publication was the base-10 lognormal. Published results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
Published 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\ \end{matrix}\,\! }[/math]
Published variance/covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 \\ {} & {} & {} \\ \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma '}}}} \right)=0.0016 \\ \end{matrix} \right]\,\! }[/math]
To replicate the published results (since Weibull++ uses a lognormal to the base [math]\displaystyle{ e\,\! }[/math] ), take the base-10 logarithm of the data and estimate the parameters using the normal distribution and MLE.
- Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
- Weibull++ computed 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\ \end{matrix}\,\! }[/math]
- Weibull++ computed/variance covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.0009 \\ {} & {} & {} \\ \widehat{Cov}({\mu }',{{{\hat{\sigma' }}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma' }}}} \right)=0.0015 \\ \end{matrix} \right]\,\! }[/math]
Interval Data Example
Determine the lognormal parameter estimates for the data given in the table below.
| Non-Grouped Data Times-to-Failure with Intervals | ||
| Data point index | Last Inspected | State End 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 |
Solution
This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.18 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.17 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.21 \end{align}\,\! }[/math]
The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units.
As may be surmised by the name, the lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Because of this, there are many mathematical similarities between the two distributions. For example, the mathematical reasoning for the construction of the probability plotting scales and the bias of parameter estimators is very similar for these two distributions.
Lognormal Probability Density Function
The lognormal distribution is a 2-parameter distribution with parameters [math]\displaystyle{ {\mu }'\,\! }[/math] and [math]\displaystyle{ \sigma'\,\! }[/math]. The pdf for this distribution is given by:
- [math]\displaystyle{ f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}\,\! }[/math]
where:
- [math]\displaystyle{ {t}'=\ln (t)\,\! }[/math]. [math]\displaystyle{ t\,\! }[/math] values are the times-to-failure
- [math]\displaystyle{ \mu'\,\! }[/math] = mean of the natural logarithms of the times-to-failure
- [math]\displaystyle{ \sigma'\,\! }[/math] = standard deviation of the natural logarithms of the times-to-failure
The lognormal pdf can be obtained, realizing that for equal probabilities under the normal and lognormal pdfs, incremental areas should also be equal, or:
- [math]\displaystyle{ \begin{align} f(t)dt=f({t}')d{t}' \end{align}\,\! }[/math]
Taking the derivative of the relationship between [math]\displaystyle{ {t}'\,\! }[/math] and [math]\displaystyle{ {t}\,\! }[/math] yields:
- [math]\displaystyle{ d{t}'=\frac{dt}{t}\,\! }[/math]
Substitution yields:
- [math]\displaystyle{ \begin{align} f(t)= & \frac{f({t}')}{t} \\ f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} \end{align}\,\! }[/math]
where:
- [math]\displaystyle{ f(t)\ge 0,t\gt 0,-\infty \lt {\mu }'\lt \infty ,{{\sigma' }}\gt 0\,\! }[/math]
Lognormal Distribution Functions
The Mean or MTTF
The mean of the lognormal distribution, [math]\displaystyle{ \mu \,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}\,\! }[/math]
The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ \mu'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {{\sigma}}\,\! }[/math] is given by:
- [math]\displaystyle{ {\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma^{2}}{{{{\bar{T}}}^{2}}}+1 \right)\,\! }[/math]
The Median
The median of the lognormal distribution, [math]\displaystyle{ \breve{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \breve{T}={{e}^{{{\mu}'}}}\,\! }[/math]
The Mode
The mode of the lognormal distribution, [math]\displaystyle{ \tilde{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \tilde{T}={{e}^{{\mu }'-\sigma'^{2}}}\,\! }[/math]
The Standard Deviation
The standard deviation of the lognormal distribution, [math]\displaystyle{ {\sigma }_{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ {\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}\,\! }[/math]
The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {\sigma}'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {\sigma}\,\! }[/math] is given by:
- [math]\displaystyle{ \sigma '=\sqrt{\ln \left( \frac{{\sigma}_{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}\,\! }[/math]
The Lognormal Reliability Function
The reliability for a mission of time [math]\displaystyle{ t\,\! }[/math], starting at age 0, for the lognormal distribution is determined by:
- [math]\displaystyle{ R(t)=\int_{t}^{\infty }f(x)dx\,\! }[/math]
or:
- [math]\displaystyle{ {{R}({t})}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
As with the normal distribution, there is no closed-form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.
The Lognormal Conditional Reliability Function
The lognormal conditional reliability function is given by:
- [math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{\text{ln}(T+t)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx}\,\! }[/math]
Once again, the use of standard normal tables is necessary to solve this equation, as no closed-form solution exists.
The Lognormal Reliable Life Function
As there is no closed-form solution for the lognormal reliability equation, no closed-form solution exists for the lognormal reliable life either. In order to determine this value, one must solve the following equation for [math]\displaystyle{ t\,\! }[/math]:
- [math]\displaystyle{ {{R}_{t}}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
The Lognormal Failure Rate Function
The lognormal failure rate is given by:
- [math]\displaystyle{ \lambda (t)=\frac{f(t)}{R(t)}=\frac{\tfrac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{t}'-{\mu }'}{{{\sigma' }}})}^{2}}}}}{\int_{{{t}'}}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{x-{\mu }'}{{{\sigma' }}})}^{2}}}}dx}\,\! }[/math]
As with the reliability equations, standard normal tables will be required to solve for this function.
Characteristics of the Lognormal Distribution
- The lognormal distribution is a distribution skewed to the right.
- The pdf starts at zero, increases to its mode, and decreases thereafter.
- The degree of skewness increases as [math]\displaystyle{ {{\sigma'}}\,\! }[/math] increases, for a given [math]\displaystyle{ \mu'\,\! }[/math]
- For the same [math]\displaystyle{ {{\sigma'}}\,\! }[/math], the pdf 's skewness increases as [math]\displaystyle{ {\mu }'\,\! }[/math] increases.
- For [math]\displaystyle{ {{\sigma' }}\,\! }[/math] values significantly greater than 1, the pdf rises very sharply in the beginning, (i.e., for very small values of [math]\displaystyle{ T\,\! }[/math] near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential pdf or a Weibull pdf with [math]\displaystyle{ 0\lt \beta \lt 1\,\! }[/math].
- The parameter, [math]\displaystyle{ {\mu }'\,\! }[/math], in terms of the logarithm of the [math]\displaystyle{ {T}'s\,\! }[/math] is also the scale parameter, and not the location parameter as in the case of the normal pdf.
- The parameter [math]\displaystyle{ {{\sigma'}}\,\! }[/math], or the standard deviation of the [math]\displaystyle{ {T}'s\,\! }[/math] in terms of their logarithm or of their [math]\displaystyle{ {T}'\,\! }[/math], is also the shape parameter and not the scale parameter, as in the normal pdf, and assumes only positive values.
Lognormal Distribution Parameters in ReliaSoft's Software
In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter [math]\displaystyle{ {\mu }'\,\! }[/math] represents the mean of the natural logarithms of the times-to-failure, while [math]\displaystyle{ {{\sigma' }}\,\! }[/math] represents the standard deviation of these data point logarithms. Specifically, the returned [math]\displaystyle{ {{\sigma' }}\,\! }[/math] is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.
Lognormal Distribution Examples
Complete Data Example
Determine the lognormal parameter estimates for the data given in the following table.
| Non-Grouped Times-to-Failure Data | ||
| Data point index | State F or S | State End Time |
|---|---|---|
| 1 | F | 2 |
| 2 | F | 5 |
| 3 | F | 11 |
| 4 | F | 23 |
| 5 | F | 29 |
| 6 | F | 37 |
| 7 | F | 43 |
| 8 | F | 59 |
Solution
Using Weibull++, the computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {\hat{\sigma '}}= & 1.10 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.24 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y:\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.36 \end{align}\,\! }[/math]
Complete Data RRX Example
From Kececioglu [20, p. 347]. 15 identical units were tested to failure and following is a table of their failure times:
Solution
Published results (using probability plotting):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.22575 \\ {{\widehat{\sigma' }}}=0.62048. \\ \end{matrix}\,\! }[/math]
Weibull++ computed parameters for rank regression on X are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.2303 \\ {{\widehat{\sigma'}}}=0.6283. \\ \end{matrix}\,\! }[/math]
The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.
Complete Data Unbiased MLE Example
From Kececioglu [19, p. 406]. 9 identical units are tested continuously to failure and failure times were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1 and 257.9 hours.
Solution
The results published were obtained by using the unbiased model. Published Results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.67677 \\ \end{matrix}\,\! }[/math]
This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results.
Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.6768 \\ \end{matrix}\,\! }[/math]
Suspension Data Example
From Nelson [30, p. 324]. 96 locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. The table below shows the failure and suspension times.
| Nelson's Locomotive Data | |||
| Number in State | F or S | Time | |
|---|---|---|---|
| 1 | 1 | F | 22.5 |
| 2 | 1 | F | 37.5 |
| 3 | 1 | F | 46 |
| 4 | 1 | F | 48.5 |
| 5 | 1 | F | 51.5 |
| 6 | 1 | F | 53 |
| 7 | 1 | F | 54.5 |
| 8 | 1 | F | 57.5 |
| 9 | 1 | F | 66.5 |
| 10 | 1 | F | 68 |
| 11 | 1 | F | 69.5 |
| 12 | 1 | F | 76.5 |
| 13 | 1 | F | 77 |
| 14 | 1 | F | 78.5 |
| 15 | 1 | F | 80 |
| 16 | 1 | F | 81.5 |
| 17 | 1 | F | 82 |
| 18 | 1 | F | 83 |
| 19 | 1 | F | 84 |
| 20 | 1 | F | 91.5 |
| 21 | 1 | F | 93.5 |
| 22 | 1 | F | 102.5 |
| 23 | 1 | F | 107 |
| 24 | 1 | F | 108.5 |
| 25 | 1 | F | 112.5 |
| 26 | 1 | F | 113.5 |
| 27 | 1 | F | 116 |
| 28 | 1 | F | 117 |
| 29 | 1 | F | 118.5 |
| 30 | 1 | F | 119 |
| 31 | 1 | F | 120 |
| 32 | 1 | F | 122.5 |
| 33 | 1 | F | 123 |
| 34 | 1 | F | 127.5 |
| 35 | 1 | F | 131 |
| 36 | 1 | F | 132.5 |
| 37 | 1 | F | 134 |
| 38 | 59 | S | 135 |
Solution
The distribution used in the publication was the base-10 lognormal. Published results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
Published 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\ \end{matrix}\,\! }[/math]
Published variance/covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 \\ {} & {} & {} \\ \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma '}}}} \right)=0.0016 \\ \end{matrix} \right]\,\! }[/math]
To replicate the published results (since Weibull++ uses a lognormal to the base [math]\displaystyle{ e\,\! }[/math] ), take the base-10 logarithm of the data and estimate the parameters using the normal distribution and MLE.
- Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
- Weibull++ computed 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\ \end{matrix}\,\! }[/math]
- Weibull++ computed/variance covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.0009 \\ {} & {} & {} \\ \widehat{Cov}({\mu }',{{{\hat{\sigma' }}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma' }}}} \right)=0.0015 \\ \end{matrix} \right]\,\! }[/math]
Interval Data Example
Determine the lognormal parameter estimates for the data given in the table below.
| Non-Grouped Data Times-to-Failure with Intervals | ||
| Data point index | Last Inspected | State End 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 |
Solution
This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.18 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.17 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.21 \end{align}\,\! }[/math]
The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units.
As may be surmised by the name, the lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Because of this, there are many mathematical similarities between the two distributions. For example, the mathematical reasoning for the construction of the probability plotting scales and the bias of parameter estimators is very similar for these two distributions.
Lognormal Probability Density Function
The lognormal distribution is a 2-parameter distribution with parameters [math]\displaystyle{ {\mu }'\,\! }[/math] and [math]\displaystyle{ \sigma'\,\! }[/math]. The pdf for this distribution is given by:
- [math]\displaystyle{ f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}\,\! }[/math]
where:
- [math]\displaystyle{ {t}'=\ln (t)\,\! }[/math]. [math]\displaystyle{ t\,\! }[/math] values are the times-to-failure
- [math]\displaystyle{ \mu'\,\! }[/math] = mean of the natural logarithms of the times-to-failure
- [math]\displaystyle{ \sigma'\,\! }[/math] = standard deviation of the natural logarithms of the times-to-failure
The lognormal pdf can be obtained, realizing that for equal probabilities under the normal and lognormal pdfs, incremental areas should also be equal, or:
- [math]\displaystyle{ \begin{align} f(t)dt=f({t}')d{t}' \end{align}\,\! }[/math]
Taking the derivative of the relationship between [math]\displaystyle{ {t}'\,\! }[/math] and [math]\displaystyle{ {t}\,\! }[/math] yields:
- [math]\displaystyle{ d{t}'=\frac{dt}{t}\,\! }[/math]
Substitution yields:
- [math]\displaystyle{ \begin{align} f(t)= & \frac{f({t}')}{t} \\ f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} \end{align}\,\! }[/math]
where:
- [math]\displaystyle{ f(t)\ge 0,t\gt 0,-\infty \lt {\mu }'\lt \infty ,{{\sigma' }}\gt 0\,\! }[/math]
Lognormal Distribution Functions
The Mean or MTTF
The mean of the lognormal distribution, [math]\displaystyle{ \mu \,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}\,\! }[/math]
The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ \mu'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {{\sigma}}\,\! }[/math] is given by:
- [math]\displaystyle{ {\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma^{2}}{{{{\bar{T}}}^{2}}}+1 \right)\,\! }[/math]
The Median
The median of the lognormal distribution, [math]\displaystyle{ \breve{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \breve{T}={{e}^{{{\mu}'}}}\,\! }[/math]
The Mode
The mode of the lognormal distribution, [math]\displaystyle{ \tilde{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \tilde{T}={{e}^{{\mu }'-\sigma'^{2}}}\,\! }[/math]
The Standard Deviation
The standard deviation of the lognormal distribution, [math]\displaystyle{ {\sigma }_{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ {\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}\,\! }[/math]
The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {\sigma}'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {\sigma}\,\! }[/math] is given by:
- [math]\displaystyle{ \sigma '=\sqrt{\ln \left( \frac{{\sigma}_{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}\,\! }[/math]
The Lognormal Reliability Function
The reliability for a mission of time [math]\displaystyle{ t\,\! }[/math], starting at age 0, for the lognormal distribution is determined by:
- [math]\displaystyle{ R(t)=\int_{t}^{\infty }f(x)dx\,\! }[/math]
or:
- [math]\displaystyle{ {{R}({t})}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
As with the normal distribution, there is no closed-form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.
The Lognormal Conditional Reliability Function
The lognormal conditional reliability function is given by:
- [math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{\text{ln}(T+t)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx}\,\! }[/math]
Once again, the use of standard normal tables is necessary to solve this equation, as no closed-form solution exists.
The Lognormal Reliable Life Function
As there is no closed-form solution for the lognormal reliability equation, no closed-form solution exists for the lognormal reliable life either. In order to determine this value, one must solve the following equation for [math]\displaystyle{ t\,\! }[/math]:
- [math]\displaystyle{ {{R}_{t}}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
The Lognormal Failure Rate Function
The lognormal failure rate is given by:
- [math]\displaystyle{ \lambda (t)=\frac{f(t)}{R(t)}=\frac{\tfrac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{t}'-{\mu }'}{{{\sigma' }}})}^{2}}}}}{\int_{{{t}'}}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{x-{\mu }'}{{{\sigma' }}})}^{2}}}}dx}\,\! }[/math]
As with the reliability equations, standard normal tables will be required to solve for this function.
Characteristics of the Lognormal Distribution
- The lognormal distribution is a distribution skewed to the right.
- The pdf starts at zero, increases to its mode, and decreases thereafter.
- The degree of skewness increases as [math]\displaystyle{ {{\sigma'}}\,\! }[/math] increases, for a given [math]\displaystyle{ \mu'\,\! }[/math]
- For the same [math]\displaystyle{ {{\sigma'}}\,\! }[/math], the pdf 's skewness increases as [math]\displaystyle{ {\mu }'\,\! }[/math] increases.
- For [math]\displaystyle{ {{\sigma' }}\,\! }[/math] values significantly greater than 1, the pdf rises very sharply in the beginning, (i.e., for very small values of [math]\displaystyle{ T\,\! }[/math] near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential pdf or a Weibull pdf with [math]\displaystyle{ 0\lt \beta \lt 1\,\! }[/math].
- The parameter, [math]\displaystyle{ {\mu }'\,\! }[/math], in terms of the logarithm of the [math]\displaystyle{ {T}'s\,\! }[/math] is also the scale parameter, and not the location parameter as in the case of the normal pdf.
- The parameter [math]\displaystyle{ {{\sigma'}}\,\! }[/math], or the standard deviation of the [math]\displaystyle{ {T}'s\,\! }[/math] in terms of their logarithm or of their [math]\displaystyle{ {T}'\,\! }[/math], is also the shape parameter and not the scale parameter, as in the normal pdf, and assumes only positive values.
Lognormal Distribution Parameters in ReliaSoft's Software
In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter [math]\displaystyle{ {\mu }'\,\! }[/math] represents the mean of the natural logarithms of the times-to-failure, while [math]\displaystyle{ {{\sigma' }}\,\! }[/math] represents the standard deviation of these data point logarithms. Specifically, the returned [math]\displaystyle{ {{\sigma' }}\,\! }[/math] is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.
Lognormal Distribution Examples
Complete Data Example
Determine the lognormal parameter estimates for the data given in the following table.
| Non-Grouped Times-to-Failure Data | ||
| Data point index | State F or S | State End Time |
|---|---|---|
| 1 | F | 2 |
| 2 | F | 5 |
| 3 | F | 11 |
| 4 | F | 23 |
| 5 | F | 29 |
| 6 | F | 37 |
| 7 | F | 43 |
| 8 | F | 59 |
Solution
Using Weibull++, the computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {\hat{\sigma '}}= & 1.10 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.24 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y:\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.36 \end{align}\,\! }[/math]
Complete Data RRX Example
From Kececioglu [20, p. 347]. 15 identical units were tested to failure and following is a table of their failure times:
Solution
Published results (using probability plotting):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.22575 \\ {{\widehat{\sigma' }}}=0.62048. \\ \end{matrix}\,\! }[/math]
Weibull++ computed parameters for rank regression on X are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.2303 \\ {{\widehat{\sigma'}}}=0.6283. \\ \end{matrix}\,\! }[/math]
The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.
Complete Data Unbiased MLE Example
From Kececioglu [19, p. 406]. 9 identical units are tested continuously to failure and failure times were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1 and 257.9 hours.
Solution
The results published were obtained by using the unbiased model. Published Results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.67677 \\ \end{matrix}\,\! }[/math]
This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results.
Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.6768 \\ \end{matrix}\,\! }[/math]
Suspension Data Example
From Nelson [30, p. 324]. 96 locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. The table below shows the failure and suspension times.
| Nelson's Locomotive Data | |||
| Number in State | F or S | Time | |
|---|---|---|---|
| 1 | 1 | F | 22.5 |
| 2 | 1 | F | 37.5 |
| 3 | 1 | F | 46 |
| 4 | 1 | F | 48.5 |
| 5 | 1 | F | 51.5 |
| 6 | 1 | F | 53 |
| 7 | 1 | F | 54.5 |
| 8 | 1 | F | 57.5 |
| 9 | 1 | F | 66.5 |
| 10 | 1 | F | 68 |
| 11 | 1 | F | 69.5 |
| 12 | 1 | F | 76.5 |
| 13 | 1 | F | 77 |
| 14 | 1 | F | 78.5 |
| 15 | 1 | F | 80 |
| 16 | 1 | F | 81.5 |
| 17 | 1 | F | 82 |
| 18 | 1 | F | 83 |
| 19 | 1 | F | 84 |
| 20 | 1 | F | 91.5 |
| 21 | 1 | F | 93.5 |
| 22 | 1 | F | 102.5 |
| 23 | 1 | F | 107 |
| 24 | 1 | F | 108.5 |
| 25 | 1 | F | 112.5 |
| 26 | 1 | F | 113.5 |
| 27 | 1 | F | 116 |
| 28 | 1 | F | 117 |
| 29 | 1 | F | 118.5 |
| 30 | 1 | F | 119 |
| 31 | 1 | F | 120 |
| 32 | 1 | F | 122.5 |
| 33 | 1 | F | 123 |
| 34 | 1 | F | 127.5 |
| 35 | 1 | F | 131 |
| 36 | 1 | F | 132.5 |
| 37 | 1 | F | 134 |
| 38 | 59 | S | 135 |
Solution
The distribution used in the publication was the base-10 lognormal. Published results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
Published 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\ \end{matrix}\,\! }[/math]
Published variance/covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 \\ {} & {} & {} \\ \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma '}}}} \right)=0.0016 \\ \end{matrix} \right]\,\! }[/math]
To replicate the published results (since Weibull++ uses a lognormal to the base [math]\displaystyle{ e\,\! }[/math] ), take the base-10 logarithm of the data and estimate the parameters using the normal distribution and MLE.
- Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
- Weibull++ computed 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\ \end{matrix}\,\! }[/math]
- Weibull++ computed/variance covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.0009 \\ {} & {} & {} \\ \widehat{Cov}({\mu }',{{{\hat{\sigma' }}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma' }}}} \right)=0.0015 \\ \end{matrix} \right]\,\! }[/math]
Interval Data Example
Determine the lognormal parameter estimates for the data given in the table below.
| Non-Grouped Data Times-to-Failure with Intervals | ||
| Data point index | Last Inspected | State End 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 |
Solution
This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.18 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.17 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.21 \end{align}\,\! }[/math]
The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units.
As may be surmised by the name, the lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Because of this, there are many mathematical similarities between the two distributions. For example, the mathematical reasoning for the construction of the probability plotting scales and the bias of parameter estimators is very similar for these two distributions.
Lognormal Probability Density Function
The lognormal distribution is a 2-parameter distribution with parameters [math]\displaystyle{ {\mu }'\,\! }[/math] and [math]\displaystyle{ \sigma'\,\! }[/math]. The pdf for this distribution is given by:
- [math]\displaystyle{ f({t}')=\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{{t}^{\prime }}-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}\,\! }[/math]
where:
- [math]\displaystyle{ {t}'=\ln (t)\,\! }[/math]. [math]\displaystyle{ t\,\! }[/math] values are the times-to-failure
- [math]\displaystyle{ \mu'\,\! }[/math] = mean of the natural logarithms of the times-to-failure
- [math]\displaystyle{ \sigma'\,\! }[/math] = standard deviation of the natural logarithms of the times-to-failure
The lognormal pdf can be obtained, realizing that for equal probabilities under the normal and lognormal pdfs, incremental areas should also be equal, or:
- [math]\displaystyle{ \begin{align} f(t)dt=f({t}')d{t}' \end{align}\,\! }[/math]
Taking the derivative of the relationship between [math]\displaystyle{ {t}'\,\! }[/math] and [math]\displaystyle{ {t}\,\! }[/math] yields:
- [math]\displaystyle{ d{t}'=\frac{dt}{t}\,\! }[/math]
Substitution yields:
- [math]\displaystyle{ \begin{align} f(t)= & \frac{f({t}')}{t} \\ f(t)= & \frac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}(t)-{\mu }'}{{{\sigma' }}} \right)}^{2}}}} \end{align}\,\! }[/math]
where:
- [math]\displaystyle{ f(t)\ge 0,t\gt 0,-\infty \lt {\mu }'\lt \infty ,{{\sigma' }}\gt 0\,\! }[/math]
Lognormal Distribution Functions
The Mean or MTTF
The mean of the lognormal distribution, [math]\displaystyle{ \mu \,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \mu ={{e}^{{\mu }'+\tfrac{1}{2}\sigma'^{2}}}\,\! }[/math]
The mean of the natural logarithms of the times-to-failure, [math]\displaystyle{ \mu'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {{\sigma}}\,\! }[/math] is given by:
- [math]\displaystyle{ {\mu }'=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma^{2}}{{{{\bar{T}}}^{2}}}+1 \right)\,\! }[/math]
The Median
The median of the lognormal distribution, [math]\displaystyle{ \breve{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \breve{T}={{e}^{{{\mu}'}}}\,\! }[/math]
The Mode
The mode of the lognormal distribution, [math]\displaystyle{ \tilde{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ \tilde{T}={{e}^{{\mu }'-\sigma'^{2}}}\,\! }[/math]
The Standard Deviation
The standard deviation of the lognormal distribution, [math]\displaystyle{ {\sigma }_{T}\,\! }[/math], is discussed in Kececioglu [19]:
- [math]\displaystyle{ {\sigma}_{T} =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)\left( {{e}^{\sigma {{'}^{2}}}}-1 \right)}\,\! }[/math]
The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {\sigma}'\,\! }[/math], in terms of [math]\displaystyle{ \bar{T}\,\! }[/math] and [math]\displaystyle{ {\sigma}\,\! }[/math] is given by:
- [math]\displaystyle{ \sigma '=\sqrt{\ln \left( \frac{{\sigma}_{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}\,\! }[/math]
The Lognormal Reliability Function
The reliability for a mission of time [math]\displaystyle{ t\,\! }[/math], starting at age 0, for the lognormal distribution is determined by:
- [math]\displaystyle{ R(t)=\int_{t}^{\infty }f(x)dx\,\! }[/math]
or:
- [math]\displaystyle{ {{R}({t})}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
As with the normal distribution, there is no closed-form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.
The Lognormal Conditional Reliability Function
The lognormal conditional reliability function is given by:
- [math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{\text{ln}(T+t)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}ds}{\int_{\text{ln}(T)}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx}\,\! }[/math]
Once again, the use of standard normal tables is necessary to solve this equation, as no closed-form solution exists.
The Lognormal Reliable Life Function
As there is no closed-form solution for the lognormal reliability equation, no closed-form solution exists for the lognormal reliable life either. In order to determine this value, one must solve the following equation for [math]\displaystyle{ t\,\! }[/math]:
- [math]\displaystyle{ {{R}_{t}}=\int_{\text{ln}(t)}^{\infty }\frac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-{\mu }'}{{{\sigma' }}} \right)}^{2}}}}dx\,\! }[/math]
The Lognormal Failure Rate Function
The lognormal failure rate is given by:
- [math]\displaystyle{ \lambda (t)=\frac{f(t)}{R(t)}=\frac{\tfrac{1}{t\cdot {{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{t}'-{\mu }'}{{{\sigma' }}})}^{2}}}}}{\int_{{{t}'}}^{\infty }\tfrac{1}{{{\sigma' }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{x-{\mu }'}{{{\sigma' }}})}^{2}}}}dx}\,\! }[/math]
As with the reliability equations, standard normal tables will be required to solve for this function.
Characteristics of the Lognormal Distribution
- The lognormal distribution is a distribution skewed to the right.
- The pdf starts at zero, increases to its mode, and decreases thereafter.
- The degree of skewness increases as [math]\displaystyle{ {{\sigma'}}\,\! }[/math] increases, for a given [math]\displaystyle{ \mu'\,\! }[/math]
- For the same [math]\displaystyle{ {{\sigma'}}\,\! }[/math], the pdf 's skewness increases as [math]\displaystyle{ {\mu }'\,\! }[/math] increases.
- For [math]\displaystyle{ {{\sigma' }}\,\! }[/math] values significantly greater than 1, the pdf rises very sharply in the beginning, (i.e., for very small values of [math]\displaystyle{ T\,\! }[/math] near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential pdf or a Weibull pdf with [math]\displaystyle{ 0\lt \beta \lt 1\,\! }[/math].
- The parameter, [math]\displaystyle{ {\mu }'\,\! }[/math], in terms of the logarithm of the [math]\displaystyle{ {T}'s\,\! }[/math] is also the scale parameter, and not the location parameter as in the case of the normal pdf.
- The parameter [math]\displaystyle{ {{\sigma'}}\,\! }[/math], or the standard deviation of the [math]\displaystyle{ {T}'s\,\! }[/math] in terms of their logarithm or of their [math]\displaystyle{ {T}'\,\! }[/math], is also the shape parameter and not the scale parameter, as in the normal pdf, and assumes only positive values.
Lognormal Distribution Parameters in ReliaSoft's Software
In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter [math]\displaystyle{ {\mu }'\,\! }[/math] represents the mean of the natural logarithms of the times-to-failure, while [math]\displaystyle{ {{\sigma' }}\,\! }[/math] represents the standard deviation of these data point logarithms. Specifically, the returned [math]\displaystyle{ {{\sigma' }}\,\! }[/math] is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.
Lognormal Distribution Examples
Complete Data Example
Determine the lognormal parameter estimates for the data given in the following table.
| Non-Grouped Times-to-Failure Data | ||
| Data point index | State F or S | State End Time |
|---|---|---|
| 1 | F | 2 |
| 2 | F | 5 |
| 3 | F | 11 |
| 4 | F | 23 |
| 5 | F | 29 |
| 6 | F | 37 |
| 7 | F | 43 |
| 8 | F | 59 |
Solution
Using Weibull++, the computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {\hat{\sigma '}}= & 1.10 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.24 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y:\,\! }[/math]
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 2.83 \\ & {{{\hat{\sigma' }}}}= & 1.36 \end{align}\,\! }[/math]
Complete Data RRX Example
From Kececioglu [20, p. 347]. 15 identical units were tested to failure and following is a table of their failure times:
Solution
Published results (using probability plotting):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.22575 \\ {{\widehat{\sigma' }}}=0.62048. \\ \end{matrix}\,\! }[/math]
Weibull++ computed parameters for rank regression on X are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=5.2303 \\ {{\widehat{\sigma'}}}=0.6283. \\ \end{matrix}\,\! }[/math]
The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.
Complete Data Unbiased MLE Example
From Kececioglu [19, p. 406]. 9 identical units are tested continuously to failure and failure times were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1 and 257.9 hours.
Solution
The results published were obtained by using the unbiased model. Published Results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.67677 \\ \end{matrix}\,\! }[/math]
This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results.
Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=4.3553 \\ {{\widehat{\sigma' }}}=0.6768 \\ \end{matrix}\,\! }[/math]
Suspension Data Example
From Nelson [30, p. 324]. 96 locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. The table below shows the failure and suspension times.
| Nelson's Locomotive Data | |||
| Number in State | F or S | Time | |
|---|---|---|---|
| 1 | 1 | F | 22.5 |
| 2 | 1 | F | 37.5 |
| 3 | 1 | F | 46 |
| 4 | 1 | F | 48.5 |
| 5 | 1 | F | 51.5 |
| 6 | 1 | F | 53 |
| 7 | 1 | F | 54.5 |
| 8 | 1 | F | 57.5 |
| 9 | 1 | F | 66.5 |
| 10 | 1 | F | 68 |
| 11 | 1 | F | 69.5 |
| 12 | 1 | F | 76.5 |
| 13 | 1 | F | 77 |
| 14 | 1 | F | 78.5 |
| 15 | 1 | F | 80 |
| 16 | 1 | F | 81.5 |
| 17 | 1 | F | 82 |
| 18 | 1 | F | 83 |
| 19 | 1 | F | 84 |
| 20 | 1 | F | 91.5 |
| 21 | 1 | F | 93.5 |
| 22 | 1 | F | 102.5 |
| 23 | 1 | F | 107 |
| 24 | 1 | F | 108.5 |
| 25 | 1 | F | 112.5 |
| 26 | 1 | F | 113.5 |
| 27 | 1 | F | 116 |
| 28 | 1 | F | 117 |
| 29 | 1 | F | 118.5 |
| 30 | 1 | F | 119 |
| 31 | 1 | F | 120 |
| 32 | 1 | F | 122.5 |
| 33 | 1 | F | 123 |
| 34 | 1 | F | 127.5 |
| 35 | 1 | F | 131 |
| 36 | 1 | F | 132.5 |
| 37 | 1 | F | 134 |
| 38 | 59 | S | 135 |
Solution
The distribution used in the publication was the base-10 lognormal. Published results (using MLE):
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
Published 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\ \end{matrix}\,\! }[/math]
Published variance/covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 \\ {} & {} & {} \\ \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma '}}}} \right)=0.0016 \\ \end{matrix} \right]\,\! }[/math]
To replicate the published results (since Weibull++ uses a lognormal to the base [math]\displaystyle{ e\,\! }[/math] ), take the base-10 logarithm of the data and estimate the parameters using the normal distribution and MLE.
- Weibull++ computed parameters for maximum likelihood are:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix}\,\! }[/math]
- Weibull++ computed 95% confidence limits on the parameters:
- [math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\ {{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\ \end{matrix}\,\! }[/math]
- Weibull++ computed/variance covariance matrix:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.0009 \\ {} & {} & {} \\ \widehat{Cov}({\mu }',{{{\hat{\sigma' }}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma' }}}} \right)=0.0015 \\ \end{matrix} \right]\,\! }[/math]
Interval Data Example
Determine the lognormal parameter estimates for the data given in the table below.
| Non-Grouped Data Times-to-Failure with Intervals | ||
| Data point index | Last Inspected | State End 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 |
Solution
This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.18 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ X\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.17 \end{align}\,\! }[/math]
For rank regression on [math]\displaystyle{ Y\ \,\! }[/math]:
- [math]\displaystyle{ \begin{align} & {{{\hat{\mu }}}^{\prime }}= & 3.64 \\ & {{{\hat{\sigma' }}}}= & 0.21 \end{align}\,\! }[/math]