The Gumbel/SEV Distribution - Revision history

Nicolette Young: /* Probability Paper */

2015-12-23T17:12:43Z

Probability Paper

← Older revision		Revision as of 17:12, 23 December 2015
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	The Gumbel probability paper resulting from this linearized ''cdf'' function is shown next.		The Gumbel probability paper resulting from this linearized ''cdf'' function is shown next.

	[[Image:WB.16 probability gumbel.png\|center\|~~350px~~\| ]]		[[Image:WB.16 probability gumbel.png\|center\|400px\| ]]

	For <math>z=0\,\!</math>, <math>t=\mu \,\!</math> and <math>R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678\,\!</math> (63.21% unreliability). For <math>z=1\,\!</math>, <math>\sigma =T-\mu \,\!</math> and <math>R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659.\,\!</math> To read <math>\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>\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>\mu \,\!</math> value.		For <math>z=0\,\!</math>, <math>t=\mu \,\!</math> and <math>R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678\,\!</math> (63.21% unreliability). For <math>z=1\,\!</math>, <math>\sigma =T-\mu \,\!</math> and <math>R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659.\,\!</math> To read <math>\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>\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>\mu \,\!</math> value.

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

2015-12-23T17:12:28Z

Characteristics of the Gumbel Distribution

← Older revision		Revision as of 17:12, 23 December 2015
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	:* As <math>\mu \,\!</math> increases, the ''pdf'' is shifted to the right.		:* As <math>\mu \,\!</math> increases, the ''pdf'' is shifted to the right.

	[[Image:WB.16 gumbel pdf.png\|center\|~~300px~~\| ]]		[[Image:WB.16 gumbel pdf.png\|center\|350px\| ]]

	:* As <math>\sigma \,\!</math> increases, the ''pdf'' spreads out and becomes shallower.		:* As <math>\sigma \,\!</math> increases, the ''pdf'' spreads out and becomes shallower.
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	:* For <math>T=\pm \infty ,\,\!</math> ''pdf''=0. For <math>T=\mu \,\!</math>, the ''pdf'' reaches its maximum point <math>\frac{1}{\sigma e}\,\!</math>		:* For <math>T=\pm \infty ,\,\!</math> ''pdf''=0. For <math>T=\mu \,\!</math>, the ''pdf'' reaches its maximum point <math>\frac{1}{\sigma e}\,\!</math>

	[[Image:WB.16 effect of sigma.png\|center\|~~300px~~\| ]]		[[Image:WB.16 effect of sigma.png\|center\|350px\| ]]

	:* The points of inflection of the ''pdf'' graph are <math>T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})\,\!</math> or <math>T\approx \mu \pm \sigma 0.96242\,\!</math>.		:* The points of inflection of the ''pdf'' graph are <math>T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})\,\!</math> or <math>T\approx \mu \pm \sigma 0.96242\,\!</math>.

Richard House at 18:01, 26 September 2012

2012-09-26T18:01:06Z

Show changes

Richard House: /* Bounds on Reliability */

2012-09-05T17:35:19Z

Bounds on Reliability

← Older revision		Revision as of 17:35, 5 September 2012
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	::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>		::<math>\widehat{z}=\frac{t-\widehat{\mu }}{\widehat{\sigma }}</math>

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

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Bounds on the Parameters */

2012-09-05T17:34:34Z

Bounds on the Parameters

← Older revision		Revision as of 17:34, 5 September 2012
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	===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:

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

	where <math>{{K}_{\alpha }}</math> is defined by:		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>		::<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.		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:		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}		::<math>\left( \begin{matrix}
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	\end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma }}^{-1}</math>		\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 [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations\|Appendix D]].		<math>\Lambda \,\!</math> is the log-likelihood function of the Gumbel distribution, described in [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations\|Appendix D]].

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

Richard House: /* Probability Paper */

2012-09-05T17:33:41Z

Probability Paper

← Older revision		Revision as of 17:33, 5 September 2012
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	[[Image:WB.16 probability gumbel.png\|center\|350px\| ]]		[[Image:WB.16 probability gumbel.png\|center\|350px\| ]]

	For <math>z=0</math> , <math>t=\mu </math> and <math>R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678</math> (63.21% unreliability). For <math>z=1</math> , <math>\sigma =T-\mu </math> and <math>R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659.</math> To read <math>\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>\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>\mu </math> value.		For <math>z=0\,\!</math> , <math>t=\mu \,\!</math> and <math>R(t)={{e}^{-{{e}^{0}}}}\approx 0.3678\,\!</math> (63.21% unreliability). For <math>z=1\,\!</math> , <math>\sigma =T-\mu \,\!</math> and <math>R(t)={{e}^{-{{e}^{1}}}}\approx 0.0659.\,\!</math> To read <math>\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>\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>\mu \,\!</math> value.

	==Confidence Bounds==		==Confidence Bounds==

Richard House: /* Gumbel Reliability Function */

2012-09-05T17:32:35Z

Gumbel Reliability Function

← Older revision		Revision as of 17:32, 5 September 2012
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	===Gumbel Reliability Function===		===Gumbel Reliability Function===
	The reliability for a mission of time <math>t</math> for the Gumbel distribution is given by:		The reliability for a mission of time <math>t\,\!</math> for the Gumbel distribution is given by:

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

Richard House: /* Characteristics of the Gumbel Distribution */

2012-09-05T17:31:28Z

Characteristics of the Gumbel Distribution

← Older revision		Revision as of 17:31, 5 September 2012
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	:* The shape of the Gumbel distribution is skewed to the left. The Gumbel ''pdf'' has no shape parameter. This means that the Gumbel ''pdf'' has only one shape, which does not change.		:* The shape of the Gumbel distribution is skewed to the left. The Gumbel ''pdf'' has no shape parameter. This means that the Gumbel ''pdf'' has only one shape, which does not change.
	:* The Gumbel ''pdf'' has location parameter <math>\mu ,</math> which is equal to the mode <math>\tilde{T},</math> but it differs from median and mean. This is because the Gumbel distribution is not symmetrical about its <math>\mu </math> .		:* The Gumbel ''pdf'' has location parameter <math>\mu ,\,\!</math> which is equal to the mode <math>\tilde{T},\,\!</math> but it differs from median and mean. This is because the Gumbel distribution is not symmetrical about its <math>\mu \,\!</math> .
	:* As <math>\mu </math> decreases, the ''pdf'' is shifted to the left.		:* As <math>\mu \,\!</math> decreases, the ''pdf'' is shifted to the left.
	:* As <math>\mu </math> increases, the ''pdf'' is shifted to the right.		:* As <math>\mu \,\!</math> increases, the ''pdf'' is shifted to the right.

	[[Image:WB.16 gumbel pdf.png\|center\|300px\| ]]		[[Image:WB.16 gumbel pdf.png\|center\|300px\| ]]

	:* As <math>\sigma </math> increases, the ''pdf'' spreads out and becomes shallower.		:* As <math>\sigma \,\!</math> increases, the ''pdf'' spreads out and becomes shallower.
	:* As <math>\sigma </math> decreases, the ''pdf'' becomes taller and narrower.		:* As <math>\sigma \,\!</math> decreases, the ''pdf'' becomes taller and narrower.
	:* For <math>T=\pm \infty ,</math> ''pdf''=0. For <math>T=\mu </math> , the ''pdf'' reaches its maximum point <math>\frac{1}{\sigma e}</math>		:* For <math>T=\pm \infty ,\,\!</math> ''pdf''=0. For <math>T=\mu \,\!</math> , the ''pdf'' reaches its maximum point <math>\frac{1}{\sigma e}\,\!</math>

	[[Image:WB.16 effect of sigma.png\|center\|300px\| ]]		[[Image:WB.16 effect of sigma.png\|center\|300px\| ]]

	:* The points of inflection of the ''pdf'' graph are <math>T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})</math> or <math>T\approx \mu \pm \sigma 0.96242</math> .		:* The points of inflection of the ''pdf'' graph are <math>T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})</math> or <math>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>{{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 =\tfrac{1}{\beta }</math> [[Appendix:_Life_Data_Analysis_References\|[32]]].		:* If times follow the Weibull distribution, then the logarithm of times follow a Gumbel distribution. 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 =\tfrac{1}{\beta }\,\!</math>, as discussed in [[Appendix:_Life_Data_Analysis_References\|[32]]].

	==Probability Paper==		==Probability Paper==

Richard House at 17:29, 5 September 2012

2012-09-05T17:29:47Z

← Older revision		Revision as of 17:29, 5 September 2012
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	{{template:LDABOOK\|16\|The Gumbel/SEV Distribution}}		{{template:LDABOOK\|16\|The Gumbel/SEV Distribution}}
	The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type I) distribution. The Gumbel distribution's ''pdf'' is skewed to the left, unlike the Weibull distribution's ''pdf'', 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) [[Appendix:_Life_Data_Analysis_References\|[27]]].		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 ''pdf'' is skewed to the left, unlike the Weibull distribution's ''pdf'', 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), as discussed in Meeker and Escobar [[Appendix:_Life_Data_Analysis_References\|[27]]].

	===Gumbel Probability Density Function===		===Gumbel Probability Density Function===

Richard House at 17:28, 5 September 2012

2012-09-05T17:28:27Z

← Older revision		Revision as of 17:28, 5 September 2012
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	{{template:LDABOOK\|16\|The Gumbel/SEV Distribution}}		{{template:LDABOOK\|16\|The Gumbel/SEV Distribution}}
	The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type I) distribution. The Gumbel distribution's ''pdf'' 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) [[Appendix: ~~Weibull References~~\|[27]]].		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 ''pdf'' is skewed to the left, unlike the Weibull distribution's ''pdf'', 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) [[Appendix:_Life_Data_Analysis_References\|[27]]].

	===Gumbel Probability Density Function===		===Gumbel Probability Density Function===
	The ~~<math>~~pdf~~</math>~~ of the Gumbel distribution is given by:		The ''pdf'' of the Gumbel distribution is given by:

	::<math>f(t)=\frac{1}{\sigma }{{e}^{z-{{e}^{z}}}}</math>		::<math>f(t)=\frac{1}{\sigma }{{e}^{z-{{e}^{z}}}}</math>
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	Some of the specific characteristics of the Gumbel distribution are the following:		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>~~pdf~~</math>~~ has no shape parameter. This means that the Gumbel ~~<math>~~pdf~~</math>~~ has only one shape, which does not change.		:* The shape of the Gumbel distribution is skewed to the left. The Gumbel ''pdf'' has no shape parameter. This means that the Gumbel ''pdf'' has only one shape, which does not change.
	:* The Gumbel ~~<math>~~pdf~~</math>~~ has location parameter <math>\mu ,</math> which is equal to the mode <math>\tilde{T},</math> but it differs from median and mean. This is because the Gumbel distribution is not symmetrical about its <math>\mu </math> .		:* The Gumbel ''pdf'' has location parameter <math>\mu ,</math> which is equal to the mode <math>\tilde{T},</math> but it differs from median and mean. This is because the Gumbel distribution is not symmetrical about its <math>\mu </math> .
	:* As <math>\mu </math> decreases, the ~~<math>~~pdf~~</math>~~ is shifted to the left.		:* As <math>\mu </math> decreases, the ''pdf'' is shifted to the left.
	:* As <math>\mu </math> increases, the ~~<math>~~pdf~~</math>~~ is shifted to the right.		:* As <math>\mu </math> increases, the ''pdf'' is shifted to the right.

	[[Image:WB.16 gumbel pdf.png\|center\|300px\| ]]		[[Image:WB.16 gumbel pdf.png\|center\|300px\| ]]

	:* As <math>\sigma </math> increases, the ~~<math>~~pdf~~</math>~~ spreads out and becomes shallower.		:* As <math>\sigma </math> increases, the ''pdf'' spreads out and becomes shallower.
	:* As <math>\sigma </math> decreases, the ~~<math>~~pdf~~</math>~~ becomes taller and narrower.		:* As <math>\sigma </math> decreases, the ''pdf'' becomes taller and narrower.
	:* For <math>T=\pm \infty ,</math> ~~<math>~~pdf=0.~~</math>~~ For <math>T=\mu </math> , the ~~<math>~~pdf~~</math>~~ reaches its maximum point <math>\frac{1}{\sigma e}</math>		:* For <math>T=\pm \infty ,</math> ''pdf''=0. For <math>T=\mu </math> , the ''pdf'' reaches its maximum point <math>\frac{1}{\sigma e}</math>

	[[Image:WB.16 effect of sigma.png\|center\|300px\| ]]		[[Image:WB.16 effect of sigma.png\|center\|300px\| ]]

	:* The points of inflection of the ~~<math>~~pdf~~</math>~~ graph are <math>T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})</math> or <math>T\approx \mu \pm \sigma 0.96242</math> .		:* The points of inflection of the ''pdf'' graph are <math>T=\mu \pm \sigma \ln (\tfrac{3\pm \sqrt{5}}{2})</math> or <math>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>{{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 =\tfrac{1}{\beta }</math> [[Appendix: ~~Weibull References~~\|[32]]].		:* If times follow the Weibull distribution, then the logarithm of times follow a Gumbel distribution. 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 =\tfrac{1}{\beta }</math> [[Appendix:_Life_Data_Analysis_References\|[32]]].

	==Probability Paper==		==Probability Paper==
	The form of the Gumbel probability paper is based on a linearization of the ~~<math>~~cdf~~</math>~~ . From the unreliabililty equation, we know:		The form of the Gumbel probability paper is based on a linearization of the ''cdf''. From the unreliabililty equation, we know:

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

	The Gumbel probability paper resulting from this linearized ~~<math>~~cdf~~</math>~~ function is shown next.		The Gumbel probability paper resulting from this linearized ''cdf'' function is shown next.

	[[Image:WB.16 probability gumbel.png\|center\|350px\| ]]		[[Image:WB.16 probability gumbel.png\|center\|350px\| ]]