Template:Gumbel/SEV Distribution - Revision history

Lisa Hacker: Redirected page to The Gumbel/SEV Distribution

2012-08-03T08:56:29Z

Redirected page to The Gumbel/SEV Distribution

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	~~{{gumbel distribution introduction}}~~		#REDIRECT [[The_Gumbel/SEV_Distribution]]

	~~{{gumbel probability density function}}~~

	~~{{gumbel mean median and mode}}~~

	~~{{gumbel standard deviation}}~~

	~~{{gumbel reliability function}}~~

	~~{{gumbel reliable life}}~~

	~~{{gumbel failure rate function}}~~

	~~{{characteristics of the gumbel distribution}}~~

	~~{{gumbel probability paper}}~~

	~~{{gumbel confidence bounds}}~~

	~~==General Examples==~~
	~~'''Example 1:'''~~
	~~{{Example: Gumbel distribution example}}~~

Harry Guo at 18:25, 20 February 2012

2012-02-20T18:25:07Z

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	{{gumbel confidence bounds}}		{{gumbel confidence bounds}}

	{{~~gumbel~~ distribution example}}		==General Examples==
			'''Example 1:'''
			{{Example: Gumbel distribution example}}

Harry Guo at 18:08, 20 February 2012

2012-02-20T18:08:19Z

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	{{gumbel confidence bounds}}		{{gumbel confidence bounds}}

			{{gumbel distribution example}}

Harry Guo at 18:06, 20 February 2012

2012-02-20T18:06:41Z

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	~~==The Gumbel/SEV Distribution==~~		{{gumbel 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>pdf</math> is skewed to the left, unlike the Weibull distribution's <math>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}}		{{gumbel probability density function}}

Nicolette Young: /* Confidence Bounds */

2012-01-04T22:51:47Z

Confidence Bounds

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	~~===Confidence Bounds===~~		{{gumbel 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.~~

Nicolette Young: /* Probability Paper */

2012-01-04T22:50:53Z

Probability Paper

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 {{characteristics of the gumbel distribution}}
-===Probability Paper===
+{{gumbel probability paper}}
 ===Confidence Bounds===

Nicolette Young: /* Characteristics of the Gumbel Distribution */

2012-01-04T22:50:16Z

Characteristics of the Gumbel Distribution

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 {{gumbel failure rate function}}
-===Characteristics of the Gumbel Distribution===
+{{characteristics of the gumbel distribution}}
 ===Probability Paper===

Nicolette Young: /* The Gumbel Failure Rate Function */

2012-01-04T22:49:31Z

The Gumbel Failure Rate Function

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	~~===The Gumbel Failure Rate Function===~~		{{gumbel failure rate function}}
	~~The instantaneous Gumbel~~ failure rate ~~is given by:~~

	~~::<math>\lambda =\frac{{{e}^{z~~}}~~}{\sigma }</math>~~


	===Characteristics of the Gumbel Distribution===		===Characteristics of the Gumbel Distribution===

Nicolette Young: /* The Gumbel Reliable Life */

2012-01-04T22:47:05Z

The Gumbel Reliable Life

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	~~===The Gumbel Reliable Life===~~		{{gumbel reliable life}}
	~~The Gumbel~~ reliable life ~~is given by:~~


	~~::<math>{{T}_{R~~}}~~=\mu +\sigma [\ln (-\ln (R))]</math>~~


	===The Gumbel Failure Rate Function===		===The Gumbel Failure Rate Function===

Nicolette Young: /* The Gumbel Reliability Function */

2012-01-04T22:46:40Z

The Gumbel Reliability Function

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	{{gumbel standard deviation}}		{{gumbel standard deviation}}

	~~===The Gumbel Reliability Function===~~		{{gumbel reliability function}}
	~~The reliability for a mission of time <math>T</math> for the Gumbel distribution is given by:~~

	~~::<math>R(T)=~~{{~~e}^{-{{e}^{z}}}}</math>~~

	~~The unreliability~~ function ~~is given by:~~

	~~::<math>F(T)=1-{{e}^{-{{e}^{z}}~~}}~~</math>~~

	===The Gumbel Reliable Life===		===The Gumbel Reliable Life===