The Generalized Gamma Distribution - Revision history

Sharon Honecker: /* Generalized Gamma Reliability Function */

2017-07-14T22:06:38Z

Generalized Gamma Reliability Function

← Older revision		Revision as of 22:06, 14 July 2017
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	::<math>R(t)=\left\{ \begin{array}{*{35}{l}}		::<math>R(t)=\left\{ \begin{array}{*{35}{l}}
	1-{{\Gamma }_{I}}\left( \tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}~~}{{{\lambda }^{2}}};\tfrac{1~~}{{{\lambda }^{2}}} \right)\text{ if }\lambda >0 \\		1-{{\Gamma }_{I}}\left( \tfrac{1}{{{\lambda }^{2}}};\tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}} \right)\text{ if }\lambda >0 \\
	1-\Phi \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)\text{ if }\lambda =0 \\		1-\Phi \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)\text{ if }\lambda =0 \\
	{{\Gamma }_{I}}\left( \tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}~~}{{{\lambda }^{2}}};\tfrac{1~~}{{{\lambda }^{2}}} \right)\text{ if }\lambda <0 \\		{{\Gamma }_{I}}\left( \tfrac{1}{{{\lambda }^{2}}};\tfrac{{{e}^{\lambda \left( \tfrac{\text{ln}(t)-\mu }{\sigma } \right)}}}{{{\lambda }^{2}}} \right)\text{ if }\lambda <0 \\
	\end{array} \right.\,\!</math>		\end{array} \right.\,\!</math>

Nicolette Young: /* Characteristics of the Generalized Gamma Distribution */

2015-12-23T17:09:33Z

Characteristics of the Generalized Gamma Distribution

← Older revision		Revision as of 17:09, 23 December 2015
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	As mentioned previously, the generalized gamma distribution includes other distributions as special cases based on the values of the parameters.		As mentioned previously, the generalized gamma distribution includes other distributions as special cases based on the values of the parameters.

	[[Image:WB chp12 pdf.png\|center\|~~350px~~\| ]]		[[Image:WB chp12 pdf.png\|center\|400px\| ]]

	:* The Weibull distribution is a special case when <math>\lambda =1\,\!</math> and:		:* The Weibull distribution is a special case when <math>\lambda =1\,\!</math> and:

Harry Guo: /* Characteristics of the Generalized Gamma Distribution */

2013-04-29T15:27:13Z

Characteristics of the Generalized Gamma Distribution

← Older revision		Revision as of 15:27, 29 April 2013
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	::<math>\begin{align}		::<math>\begin{align}
	& \beta = \frac{1}{\sigma } \\		& \beta = \frac{1}{\sigma } \\
	& \eta = \ln (\mu )		& \eta = \exp (\mu )
	\end{align}\,\!</math>		\end{align}\,\!</math>

Richard House at 16:13, 26 September 2012

2012-09-26T16:13:07Z

Show changes

Richard House: /* Generalized Gamma Probability Density Function */

2012-09-05T16:39:37Z

Generalized Gamma Probability Density Function

← Older revision		Revision as of 16:39, 5 September 2012
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	::<math>\Gamma (x)=\int_{0}^{\infty }{{{s}^{x-1}}}\cdot {{e}^{-s}}ds</math>		::<math>\Gamma (x)=\int_{0}^{\infty }{{{s}^{x-1}}}\cdot {{e}^{-s}}ds</math>

	With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of <math>k\,\!</math>, <math>\beta\,\!</math>, and <math>\theta \,\!</math> may appear almost identical [[Appendix: ~~Weibull References~~\|[21]]]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters <math>k\,\!</math>, <math>\beta\,\!</math>, and <math>\theta \,\!</math> [[Appendix: ~~Weibull References~~\|[21]]] where:		With this version of the distribution, however, convergence problems arise that severely limit its usefulness. Even with data sets containing 200 or more data points, the MLE methods may fail to converge. Further adding to the confusion is the fact that distributions with widely different values of <math>k\,\!</math>, <math>\beta\,\!</math>, and <math>\theta \,\!</math> may appear almost identical, as discussed in Lawless [[Appendix:_Life_Data_Analysis_References\|[21]]]. In order to overcome these difficulties, Weibull++ uses a reparameterization with parameters <math>k\,\!</math>, <math>\beta\,\!</math>, and <math>\theta \,\!</math>, as shown in [[Appendix:_Life_Data_Analysis_References\|[21]]], where:

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

Richard House: /* Generalized Gamma Failure Rate Function */

2012-09-04T22:58:53Z

Generalized Gamma Failure Rate Function

← Older revision		Revision as of 22:58, 4 September 2012
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	::<math>\lambda (t)=\frac{f(t)}{R(t)}</math>		::<math>\lambda (t)=\frac{f(t)}{R(t)}</math>

	Owing to the complexity of the equations involved, the function will not be displayed here, but the failure rate function for the generalized gamma distribution can be obtained merely by dividing the 'pdf' function by the reliability function.		Owing to the complexity of the equations involved, the function will not be displayed here, but the failure rate function for the generalized gamma distribution can be obtained merely by dividing the ''pdf'' function by the reliability function.

	===Generalized Gamma Reliable Life===		===Generalized Gamma Reliable Life===

Richard House: /* Bounds on the Parameters */

2012-08-28T03:18:57Z

Bounds on the Parameters

← Older revision		Revision as of 03:18, 28 August 2012
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	===Bounds on the Parameters===		===Bounds on the Parameters===
	The lower and upper bounds on the parameter <math>\mu </math> are estimated from:		The lower and upper bounds on the parameter <math>\mu \,\!</math> are estimated from:

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

Richard House: /* Bounds on Time */

2012-08-28T03:17:01Z

Bounds on Time

← Older revision		Revision as of 03:17, 28 August 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. 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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	\end{align}</math>		\end{align}</math>

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

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

	The variance of <math>u</math> is estimated from:		The variance of <math>u\,\!</math> is estimated from:

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

Richard House: /* Bounds on the Parameters */

2012-08-28T03:16:28Z

Bounds on the Parameters

← Older revision		Revision as of 03:16, 28 August 2012
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	\end{align}</math>		\end{align}</math>

	For the parameter <math>\lambda ,</math> the bounds are estimated from:		For the parameter <math>\lambda ,\,\!</math> 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 as follows:		The variances and covariances of <math>\widehat{\mu }\,\!</math> and <math>\widehat{\sigma }\,\!</math> are estimated as follows:

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

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

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

Richard House: /* Characteristics of the Generalized Gamma Distribution */

2012-08-28T03:15:47Z

Characteristics of the Generalized Gamma Distribution

← Older revision		Revision as of 03:15, 28 August 2012
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	[[Image:WB chp12 pdf.png\|center\|350px\| ]]		[[Image:WB chp12 pdf.png\|center\|350px\| ]]

	:* The Weibull distribution is a special case when <math>\lambda =1</math> and:		:* The Weibull distribution is a special case when <math>\lambda =1\,\!</math> and:

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

	:* In this case, the generalized distribution has the same behavior as the Weibull for <math>\sigma >1,</math> <math>\sigma =1,</math> and <math>\sigma <1</math> ( <math>\beta		:* In this case, the generalized distribution has the same behavior as the Weibull for <math>\sigma >1,\,\!</math> <math>\sigma =1,\,\!</math> and <math>\sigma <1\,\!</math> ( <math>\beta
	<1,</math> <math>\beta =1,</math> and <math>\beta >1</math> respectively).		<1,\,\!</math> <math>\beta =1,\,\!</math> and <math>\beta >1\,\!</math> respectively).
	:* The exponential distribution is a special case when <math>\lambda =1</math> and <math>\sigma =1</math>.		:* The exponential distribution is a special case when <math>\lambda =1\,\!</math> and <math>\sigma =1\,\!</math>.
	:* The lognormal distribution is a special case when <math>\lambda =0</math>.		:* The lognormal distribution is a special case when <math>\lambda =0\,\!</math>.
	:* The gamma distribution is a special case when <math>\lambda =\sigma </math>.		:* The gamma distribution is a special case when <math>\lambda =\sigma \,\!</math>.

	By allowing <math>\lambda </math> to take negative values, the generalized gamma distribution can be further extended to include additional distributions as special cases. For example, the Fréchet distribution of maxima (also known as a reciprocal Weibull) is a special case when <math>\lambda =-1</math>.		By allowing <math>\lambda \,\!</math> to take negative values, the generalized gamma distribution can be further extended to include additional distributions as special cases. For example, the Fréchet distribution of maxima (also known as a reciprocal Weibull) is a special case when <math>\lambda =-1\,\!</math>.

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