Template:Gamma distribution confidence bounds - Revision history

Lisa Hacker: Redirected page to The Gamma Distribution

2012-08-03T08:58:39Z

Redirected page to The Gamma Distribution

← Older revision		Revision as of 08:58, 3 August 2012
Line 1:		Line 1:
	~~===Confidence Bounds===~~		#REDIRECT [[The_Gamma_Distribution]]
	The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in chapter [[~~Confidence Bounds~~]].

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

	~~::<math>\begin{align}~~
	~~& {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{{\hat{k}}}}}\text{ (upper bound)} \\~~
	~~& {{k}_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{\widehat{k}}}}}\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{k}</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{k} \right) \\~~
	~~\widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) & \widehat{Var}\left( \widehat{k} \right) \\~~
	~~\end{matrix} \right)=\left( \begin{matrix}~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} \\~~
	~~{} & {} \\~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}} \\~~
	~~\end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1}</math>~~


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

	~~====Bounds on Reliability====~~
	~~The reliability of the gamma distribution is:~~

	~~::<math>\widehat{R}(t;\hat{\mu },\hat{k})=1-{{\Gamma }_{I}}(\widehat{k};{{e}^{\widehat{z}}})</math>~~

	~~where:~~

	~~::<math>\widehat{z}=\ln (t)-\widehat{\mu }</math>~~

	~~The upper and lower bounds on reliability are:~~

	~~::<math>{{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (upper bound)}</math>~~

	~~::<math>{{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (lower bound)}</math>~~

	~~where:~~

	::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>

	~~====Bounds on Time====~~
	~~The bounds around time for a given gamma 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 at 22:30, 14 February 2012

2012-02-14T22:30:35Z

← Older revision		Revision as of 22:30, 14 February 2012
Line 37:		Line 37:

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


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

Harry Guo at 22:30, 14 February 2012

2012-02-14T22:30:20Z

← Older revision		Revision as of 22:30, 14 February 2012
Line 37:		Line 37:

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


	====Bounds on Reliability====		====Bounds on Reliability====
	The reliability of the gamma distribution is:		The reliability of the gamma distribution is:

	::<math>\widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\widehat{k};{{e}^{\widehat{z}}})</math>		::<math>\widehat{R}(t;\hat{\mu },\hat{k})=1-{{\Gamma }_{I}}(\widehat{k};{{e}^{\widehat{z}}})</math>

	:where:		where:

	::<math>\widehat{z}=\ln (t)-\widehat{\mu }</math>		::<math>\widehat{z}=\ln (t)-\widehat{\mu }</math>
Line 52:		Line 54:
	::<math>{{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (lower bound)}</math>		::<math>{{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (lower bound)}</math>

	:where:		where:

	::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>		::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>
Line 61:		Line 63:
	::<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>
Line 67:		Line 69:
	::<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 at 22:28, 14 February 2012

2012-02-14T22:28:43Z

← Older revision		Revision as of 22:28, 14 February 2012
Line 1:		Line 1:
	===Confidence Bounds===		===Confidence Bounds===
	The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in ~~Chapter~~ [[Confidence Bounds]].		The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in chapter [[Confidence Bounds]].

	====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:
Line 35:		Line 36:


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

Harry Guo: /* Confidence Bounds */

2012-02-14T22:26:51Z

Confidence Bounds

← Older revision		Revision as of 22:26, 14 February 2012
Line 1:		Line 1:
	===Confidence Bounds===		===Confidence Bounds===
	The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in Chapter 5.		The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in Chapter [[Confidence Bounds]].
	====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:

Harry Guo: moved Template:Gd confidence bounds to Template:Gamma distribution confidence bounds

2012-02-14T22:25:23Z

moved Template:Gd confidence bounds to Template:Gamma distribution confidence bounds

← Older revision	Revision as of 22:25, 14 February 2012
(No difference)

Harry Guo at 22:05, 14 February 2012

2012-02-14T22:05:48Z

← Older revision		Revision as of 22:05, 14 February 2012
Line 54:		Line 54:

	::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>		::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>

			====Bounds on Time====
			The bounds around time for a given gamma 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 Reliability */

2012-02-14T22:03:04Z

Bounds on Reliability

← Older revision		Revision as of 22:03, 14 February 2012
Line 36:		Line 36:

	<math>\Lambda </math> is the log-likelihood function of the gamma distribution, described in Chapter 3 and Appendix C.		<math>\Lambda </math> is the log-likelihood function of the gamma distribution, described in Chapter 3 and Appendix C.
	===Bounds on Reliability===		====Bounds on Reliability====
	The reliability of the gamma distribution is:		The reliability of the gamma distribution is:

Nicolette Young: /* Confidence Bounds */

2012-01-04T21:27:16Z

Confidence Bounds

← Older revision		Revision as of 21:27, 4 January 2012
Line 36:		Line 36:

	<math>\Lambda </math> is the log-likelihood function of the gamma distribution, described in Chapter 3 and Appendix C.		<math>\Lambda </math> is the log-likelihood function of the gamma distribution, described in Chapter 3 and Appendix C.
			===Bounds on Reliability===
			The reliability of the gamma distribution is:

			::<math>\widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\widehat{k};{{e}^{\widehat{z}}})</math>

			:where:

			::<math>\widehat{z}=\ln (t)-\widehat{\mu }</math>

			The upper and lower bounds on reliability are:

			::<math>{{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (upper bound)}</math>

			::<math>{{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (lower bound)}</math>

			:where:

			::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>

Nicolette Young: Created page with '===Confidence Bounds=== The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete deriva…'

2012-01-04T21:26:47Z

Created page with '===Confidence Bounds=== The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete deriva…'

New page

===Confidence Bounds===
The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in Chapter 5.
====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, <math>\ln (\widehat{\sigma })</math> is treated as normally distributed and the bounds are estimated from:

::<math>\begin{align}
& {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{{\hat{k}}}}}\text{ (upper bound)} \\
& {{k}_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{\widehat{k}}}}}\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{k}</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{k} \right) \\
\widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) & \widehat{Var}\left( \widehat{k} \right) \\
\end{matrix} \right)=\left( \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} \\
{} & {} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}} \\
\end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1}</math>

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