The Gumbel/SEV Distribution

Introduction

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{ \begin{align} {{t}_{R}}=\mu +\sigma [\ln (-\ln (R))] \end{align} }[/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].

Probability Paper

The form of the Gumbel probability paper is based on a linearization of the [math]\displaystyle{ cdf }[/math] . From the unreliabililty equation, we know:

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

using the equation for z, we get:

[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 Confidence Bounds. 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 Parameter Estimation and Appendix D.

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]

General Examples

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. The following picture shows the Main tab of the Monte Carlo Data Generation utility.

On the Settings tab, set the number of points to 100 and click Generate. This creates a new data sheet in the folio that contains random time values [math]\displaystyle{ {{t}_{i}}\,\! }[/math].

Insert a new data sheet in the folio and enter the corresponding [math]\displaystyle{ Ln({{t}_{i}})\,\! }[/math] values of the time values generated by the Monte Carlo simulation. Delete any negative values, if there are any, because Weibull++ expects the time values to be positive. After obtaining the [math]\displaystyle{ Ln({{t}_{i}})\,\! }[/math] values, analyze the data sheet using the Gumbel distribution and the MLE parameter estimation method. The estimated parameters are (your results may vary due to the random numbers generated by simulation):

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

Because [math]\displaystyle{ \ln (\eta )= 9.2103\,\! }[/math] ( [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.