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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, 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:
 
::<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>
 
====A Generalized Gamma Distribution Example====
The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [21].
 
 
::<math>\begin{array}{*{35}{l}}
  \text{17}\text{.88} & \text{28}\text{.92} & \text{33} & \text{41}\text{.52} & \text{42}\text{.12} & \text{45}\text{.6} & \text{48}\text{.4} & \text{51}\text{.84} & \text{51}\text{.96} & \text{54}\text{.12}  \\
  \text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84}  \\
  \text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {}  \\
\end{array}</math>
 
When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:
 
::<math>\begin{align}
  & \widehat{\mu }= & 4.23064 \\
& \widehat{\sigma }= & 0.509982 \\
& \widehat{\lambda }= & 0.307639 
\end{align}</math>
 
Note that for this data, the generalized gamma offers a compromise between the Weibull  <math>(\lambda =1),</math>  and the lognormal  <math>(\lambda =0)</math>  distributions. The value of  <math>\lambda </math>  indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on  <math>\lambda .</math>  For example, the 90% two-sided confidence bounds are:
 
::<math>\begin{align}
  & {{\lambda }_{u}}= & -0.592087 \\
& {{\lambda }_{u}}= & 1.20736 
\end{align}</math>
 
It can be then concluded that both distributions (i.e. Weibull and lognormal) are well supported by the data, with the lognormal being the ,better supported of the two.
In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next.
 
[[Image:ldagamma10.2.gif|thumb|center|500px| ]]  
 
It is important to also note that as in the case of the mixed Weibull distribution, in the case of regression analysis, using a generalized gamma model, the choice of regression axis, i.e.  <math>RRX</math>  or  <math>RRY,</math>  is of no consequence since non-linear regression is utilized.

Latest revision as of 09:36, 9 August 2012

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