Template:Gumbel confidence bounds - Revision history

Lisa Hacker: Redirected page to The Gumbel/SEV Distribution

2012-08-03T08:56:51Z

Redirected page to The Gumbel/SEV Distribution

← Older revision		Revision as of 08:56, 3 August 2012
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	~~==Confidence Bounds==~~		#REDIRECT [[The_Gumbel/SEV_Distribution]]
	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 [[~~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>\widehat{\mu }</math> , are estimated from:~~

	~~::<math>\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>\widehat{\sigma }</math> , must be positive, then <math>\ln (\widehat{\sigma })</math> is treated as normally distributed, and the bounds are estimated from:~~

	~~::<math>\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>{{K}_{\alpha }}</math> is defined by:~~

	~~::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>~~


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

	~~The variances and covariances of <math>\widehat{\mu }</math> and <math>\widehat{\sigma }</math> are estimated from the Fisher matrix as follows:~~

	~~::<math>\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>\Lambda </math> is the log-likelihood function of the Gumbel distribution, described in chapter [[Parameter Estimation~~]] ~~and [[Appendix: Distribution Log-Likelihood Equations]].~~

	~~===Bounds on Reliability===~~
	~~The reliability of the Gumbel distribution is given by:~~

	~~::<math>\widehat{R}(t;\hat{\mu },\hat{\sigma })={{e}^{-{{e}^{{\hat{z}}}}}}</math>~~

	~~where:~~

	~~::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>~~

	~~The bounds on <math>z</math> are estimated from:~~

	~~::<math>\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>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>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>\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>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>~~

	~~where:~~

	~~::<math>z=\ln (-\ln (R))</math>~~

	::<math>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>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>\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>~~

Harry Guo: /* Bounds on Time */

2012-02-20T18:39:54Z

Bounds on Time

← Older revision		Revision as of 18:39, 20 February 2012
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	::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>		::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>

	:where:		where:

	::<math>z=\ln (-\ln (R))</math>		::<math>z=\ln (-\ln (R))</math>
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	::<math>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>		::<math>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:		or:

	::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>		::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>

Harry Guo: /* Bounds on Reliability */

2012-02-20T18:39:24Z

Bounds on Reliability

← Older revision		Revision as of 18:39, 20 February 2012
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	The reliability of the Gumbel distribution is given by:		The reliability of the Gumbel distribution is given by:

	::<math>\widehat{R}(T;\hat{\mu },\hat{\sigma })={{e}^{-{{e}^{{\hat{z}}}}}}</math>		::<math>\widehat{R}(t;\hat{\mu },\hat{\sigma })={{e}^{-{{e}^{{\hat{z}}}}}}</math>

	:where:		where:

	::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>		::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>
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	\end{align}</math>		\end{align}</math>

	:where:		where:

	::<math>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>		::<math>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:		or:

	::<math>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>		::<math>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>

Harry Guo: /* Confidence Bounds */

2012-02-20T18:38:40Z

Confidence Bounds

← Older revision		Revision as of 18:38, 20 February 2012
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	==Confidence Bounds==		==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~~ [[Confidence Bounds]]. Only Fisher Matrix confidence bounds are available for the Gumbel distribution.		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 [[Confidence Bounds]]. Only Fisher Matrix confidence bounds are available for the Gumbel distribution.

	===Bounds on the Parameters===		===Bounds on the Parameters===
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	<math>\Lambda </math> is the log-likelihood function of the Gumbel distribution, described in ~~Chapter 3~~ and Appendix C.		<math>\Lambda </math> is the log-likelihood function of the Gumbel distribution, described in chapter [[Parameter Estimation]] and [[Appendix: Distribution Log-Likelihood Equations]].

	===Bounds on Reliability===		===Bounds on Reliability===

Harry Guo: /* Confidence Bounds */

2012-02-20T18:37:12Z

Confidence Bounds

← Older revision		Revision as of 18:37, 20 February 2012
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	==Confidence Bounds==		==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.		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 [[Confidence Bounds]]. Only Fisher Matrix confidence bounds are available for the Gumbel distribution.

	===Bounds on the Parameters===		===Bounds on the Parameters===

Harry Guo at 18:08, 20 February 2012

2012-02-20T18:08:13Z

← Older revision		Revision as of 18:08, 20 February 2012
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	===Confidence Bounds===		==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.		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====		===Bounds on the Parameters===
	The lower and upper bounds on the mean, <math>\widehat{\mu }</math> , are estimated from:		The lower and upper bounds on the mean, <math>\widehat{\mu }</math> , are estimated from:

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	<math>\Lambda </math> is the log-likelihood function of the Gumbel distribution, described in Chapter 3 and Appendix C.		<math>\Lambda </math> is the log-likelihood function of the Gumbel distribution, described in Chapter 3 and Appendix C.

	====Bounds on Reliability====		===Bounds on Reliability===
	The reliability of the Gumbel distribution is given by:		The reliability of the Gumbel distribution is given by:

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	\end{align}</math>		\end{align}</math>

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

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	& {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}		& {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}
	\end{align}</math>		\end{align}</math>

	~~{{gumbel distribution example}}~~

Nicolette Young: /* A Gumbel Distribution Example */

2012-01-19T22:35:01Z

A Gumbel Distribution Example

← Older revision		Revision as of 22:35, 19 January 2012
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	\end{align}</math>		\end{align}</math>

	~~====A Gumbel Distribution Example====~~		{{gumbel distribution example}}
	~~Verify using Monte Carlo simulation that if <math>~~{{~~t}_{i}}</math> follows a Weibull~~ distribution with <math>\beta </math> and <math>\eta </math> , then the <math>Ln({{t}_{i}})</math> follows a Gumbel distribution with <math>\mu =\ln (\eta )</math> and <math>\sigma =1/\beta ).</math>

	Let us assume that <math>{{t}_{i}}</math> follows a Weibull distribution with <math>\beta =0.5</math> and <math>\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.

	~~[[Image:montecarlo4eva.png\|thumb\|center\|400px\| ]]~~

	After obtaining the random time values <math>{{t}_{i}}</math> , insert a new Data Sheet using the Insert Data Sheet option under the Folio menu. In this sheet enter the <math>Ln({{t}_{i}})</math> values using the LN function and referring to the cells in the sheet that contains the <math>{{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>Ln({{t}_{i}})</math> values.

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

	~~::<math>\begin{align}~~
	~~& \hat{\mu }= & 9.3816 \\~~
	~~& \hat{\sigma~~ }~~= & 1.9717~~
	~~\end{align~~}~~</math>~~


	Since <math>\ln (\eta )=</math> 9.2103 ( <math>\simeq 9.3816</math> ) and <math>1/\beta =2</math> <math>(\simeq 1.9717),</math> then this simulation verifies that <math>Ln({{t}_{i}})</math> follows a Gumbel distribution with <math>\mu =\ln (\eta )</math> and <math>\delta =1/\beta .</math>

	~~Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.~~

Nicolette Young: Created page with '===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…'

2012-01-04T22:51:55Z

Created page with '===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…'

New page

===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>\widehat{\mu }</math> , are estimated from:

::<math>\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>\widehat{\sigma }</math> , must be positive, then <math>\ln (\widehat{\sigma })</math> is treated as normally distributed, and the bounds are estimated from:

::<math>\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>{{K}_{\alpha }}</math> is defined by:

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

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

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

::<math>\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>\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>\widehat{R}(T;\hat{\mu },\hat{\sigma })={{e}^{-{{e}^{{\hat{z}}}}}}</math>

:where:

::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>

The bounds on <math>z</math> are estimated from:

::<math>\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>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>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>\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>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>

:where:

::<math>z=\ln (-\ln (R))</math>

::<math>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>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>\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>{{t}_{i}}</math> follows a Weibull distribution with <math>\beta </math> and <math>\eta </math> , then the <math>Ln({{t}_{i}})</math> follows a Gumbel distribution with <math>\mu =\ln (\eta )</math> and <math>\sigma =1/\beta ).</math>

Let us assume that <math>{{t}_{i}}</math> follows a Weibull distribution with <math>\beta =0.5</math> and <math>\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.

[[Image:montecarlo4eva.png|thumb|center|400px| ]]

After obtaining the random time values <math>{{t}_{i}}</math> , insert a new Data Sheet using the Insert Data Sheet option under the Folio menu. In this sheet enter the <math>Ln({{t}_{i}})</math> values using the LN function and referring to the cells in the sheet that contains the <math>{{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>Ln({{t}_{i}})</math> values.

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

::<math>\begin{align}
& \hat{\mu }= & 9.3816 \\
& \hat{\sigma }= & 1.9717
\end{align}</math>

Since <math>\ln (\eta )=</math> 9.2103 ( <math>\simeq 9.3816</math> ) and <math>1/\beta =2</math> <math>(\simeq 1.9717),</math> then this simulation verifies that <math>Ln({{t}_{i}})</math> follows a Gumbel distribution with <math>\mu =\ln (\eta )</math> and <math>\delta =1/\beta .</math>

Note: This example illustrates a property of the Gumbel distribution; it is not meant to be a formal proof.