Template:Generalized Gamma Confidence bounds - Revision history

Lisa Hacker: Redirected page to The Generalized Gamma Distribution

2012-08-09T09:36:01Z

Redirected page to The Generalized Gamma Distribution

← Older revision		Revision as of 09:36, 9 August 2012
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	~~===Confidence Bounds===~~		#REDIRECT [[The_Generalized_Gamma_Distribution]]
	~~The only method available in Weibull++ for confidence bounds for the generalized gamma distribution is the Fisher matrix, which is described next.~~

	~~====Bounds on the Parameters====~~
	~~The lower and upper bounds on the parameter <math>\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>~~

	~~For the parameter <math>\widehat{\sigma }</math> , <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 }}}}\text{ (upper bound)} \\~~
	~~& {{\sigma }_{L}}= \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}}\text{ (lower bound)}~~
	~~\end{align}</math>~~


	~~For the parameter <math>\lambda ,</math> the bounds are estimated from:~~

	~~::<math>\begin{align}~~
	~~& {{\lambda }_{U}}= & \widehat{\lambda }+{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\text{ (upper bound)} \\~~
	~~& {{\lambda }_{L}}= & \widehat{\lambda }-{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\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 as follows:~~


	~~::<math>\begin{align}~~
	~~\left( \begin{matrix}~~
	~~\widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\lambda } \right) \\~~
	~~\widehat{Cov}\left( \widehat{\sigma },\widehat{\mu } \right) & \widehat{Var}\left( \widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\sigma },\widehat{\lambda } \right) \\~~
	~~\widehat{Cov}\left( \widehat{\lambda },\widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\lambda },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\lambda } \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 \lambda } \\~~
	-\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } \\
	-\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \lambda } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \\
	~~\end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma },\lambda =\hat{\lambda }}^{-1}~~
	~~\end{align}</math>~~

	~~Where <math>\Lambda </math> is the log-likelihood function of the generalized gamma distribution.~~

	~~====Bounds on Reliability====~~
	~~The upper and lower bounds on reliability are given by:~~

	~~::<math>\begin{align}~~
	~~& {{R}_{U}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}} \\~~
	~~& {{R}_{L}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}}~~
	~~\end{align}</math>~~

	~~where:~~

	~~::<math>\begin{align}~~
	Var(\widehat{R})= & {{\left( \frac{\partial R}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial R}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	& +2\left( \frac{\partial R}{\partial \mu } \right)\left( \frac{\partial R}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial R}{\partial \mu } \right)\left( \frac{\partial R}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	~~& +2\left( \frac{\partial R}{\partial \lambda } \right)\left( \frac{\partial R}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })~~
	~~\end{align}</math>~~

	~~====Bounds on Time====~~
	The bounds around time for a given percentile, or unreliability, are estimated by first solving the reliability equation with respect to time. Since <math>T</math> is a positive variable, the transformed variable <math>\hat{u}=\ln (\widehat{T})</math> is treated as normally distributed and the bounds are estimated from:

	~~::<math>\begin{align}~~
	~~& {{u}_{u}}= & \ln {{T}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} \\~~
	~~& {{u}_{L}}= & \ln {{T}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}~~
	~~\end{align}</math>~~

	~~Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> we get:~~

	~~::<math>\begin{align}~~
	~~& {{T}_{U}}= & {{e}^{{{T}_{U}}}}\text{ (upper bound)} \\~~
	~~& {{T}_{L}}= & {{e}^{{{T}_{L}}}}\text{ (lower bound)}~~
	~~\end{align}</math>~~

	~~The variance of <math>u</math> is estimated from:~~

	~~::<math>\begin{align}~~
	& Var(\widehat{u})= {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	& +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	~~& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })~~
	~~\end{align}</math>~~

Harry Guo: /* Confidence Bounds */

2012-02-14T22:08:38Z

Confidence Bounds

← Older revision		Revision as of 22:08, 14 February 2012
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	& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
	\end{align}</math>		\end{align}</math>


	~~'''Example 1:'''~~
	~~{{Example: Generalized Gamma Distribution Example}}~~

Harry Guo: /* Bounds on Time */

2012-02-14T21:56:10Z

Bounds on Time

← Older revision		Revision as of 21:56, 14 February 2012
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	& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
	\end{align}</math>		\end{align}</math>


	'''Example 1:'''		'''Example 1:'''
	{{Example: Generalized Gamma Distribution Example}}		{{Example: Generalized Gamma Distribution Example}}

Harry Guo: /* Bounds on Time */

2012-02-14T21:56:01Z

Bounds on Time

← Older revision		Revision as of 21:56, 14 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{u})= {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\		& Var(\widehat{u})= {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	& & +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\		& +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	& & +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
	\end{align}</math>		\end{align}</math>

	'''Example 1:'''		'''Example 1:'''
	{{Example: Generalized Gamma Distribution Example}}		{{Example: Generalized Gamma Distribution Example}}

Harry Guo: /* Bounds on Time */

2012-02-14T21:55:47Z

Bounds on Time

← Older revision		Revision as of 21:55, 14 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{u})= {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\		& Var(\widehat{u})= {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	& +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\		& & +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		& & +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
	\end{align}</math>		\end{align}</math>

	'''Example 1:'''		'''Example 1:'''
	{{Example: Generalized Gamma Distribution Example}}		{{Example: Generalized Gamma Distribution Example}}

Harry Guo: /* Bounds on Time */

2012-02-14T21:55:25Z

Bounds on Time

← Older revision		Revision as of 21:55, 14 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\		& Var(\widehat{u})= {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	& +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\		& +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })

Harry Guo: /* Bounds on Time */

2012-02-14T21:55:09Z

Bounds on Time

← Older revision		Revision as of 21:55, 14 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\		& Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\		& +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	+2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		& +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
	\end{align}</math>		\end{align}</math>

	'''Example 1:'''		'''Example 1:'''
	{{Example: Generalized Gamma Distribution Example}}		{{Example: Generalized Gamma Distribution Example}}

Harry Guo: /* Bounds on Time */

2012-02-14T21:54:24Z

Bounds on Time

← Older revision		Revision as of 21:54, 14 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\		Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\		+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	+2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		+2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })

Harry Guo: /* Bounds on Time */

2012-02-14T21:54:06Z

Bounds on Time

← Older revision		Revision as of 21:54, 14 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\		& Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\		+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	+2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		+2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })

Harry Guo: /* Bounds on Time */

2012-02-14T21:53:42Z

Bounds on Time

← Older revision		Revision as of 21:53, 14 February 2012
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	====Bounds on Time====		====Bounds on Time====
	The bounds around time for a given percentile, or unreliability, are estimated by first solving the reliability equation with respect to time~~, given by Eqn. (GGamma Time)~~. Since <math>T</math> is a positive variable, the transformed variable <math>\hat{u}=\ln (\widehat{T})</math> is treated as normally distributed and the bounds are estimated from:		The bounds around time for a given percentile, or unreliability, are estimated by first solving the reliability equation with respect to time. Since <math>T</math> is a positive variable, the transformed variable <math>\hat{u}=\ln (\widehat{T})</math> is treated as normally distributed and the bounds are estimated from:

	::<math>\begin{align}		::<math>\begin{align}
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	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+ \\		Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })\\
	~~& &~~ +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })+ \\		+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })\\
	~~& &~~ +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })		+2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
	\end{align}</math>		\end{align}</math>

	'''Example 1:'''		'''Example 1:'''
	{{Example: Generalized Gamma Distribution Example}}		{{Example: Generalized Gamma Distribution Example}}