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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.
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| ====Bounds on the Parameters====
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| The lower and upper bounds on the parameter <math>\mu </math> are estimated from:
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| ::<math>\begin{align}
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| & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\
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| & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)}
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| \end{align}</math>
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| For the parameter <math>\widehat{\sigma }</math> , <math>\ln (\widehat{\sigma })</math> is treated as normally distributed, and the bounds are estimated from:
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| ::<math>\begin{align}
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| & {{\sigma }_{U}}= & \widehat{\sigma }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}\text{ (upper bound)} \\
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| & {{\sigma }_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}}\text{ (lower bound)}
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| \end{align}</math>
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| For the parameter <math>\lambda ,</math> the bounds are estimated from:
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| ::<math>\begin{align}
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| & {{\lambda }_{U}}= & \widehat{\lambda }+{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\text{ (upper bound)} \\
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| & {{\lambda }_{L}}= & \widehat{\lambda }-{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\text{ (lower bound)}
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| \end{align}</math>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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|
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| 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.
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| The variances and covariances of <math>\widehat{\mu }</math> and <math>\widehat{\sigma }</math> are estimated as follows:
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| ::<math>\begin{align}
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| & & \left( \begin{matrix}
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| \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\lambda } \right) \\
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| \widehat{Cov}\left( \widehat{\sigma },\widehat{\mu } \right) & \widehat{Var}\left( \widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\sigma },\widehat{\lambda } \right) \\
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| \widehat{Cov}\left( \widehat{\lambda },\widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\lambda },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\lambda } \right) \\
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| \end{matrix} \right) \\
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| & = & \left( \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \lambda } \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \lambda } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \\
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| \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma },\lambda =\hat{\lambda }}^{-1}
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| \end{align}</math>
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| Where <math>\Lambda </math> is the log-likelihood function of the generalized gamma distribution.
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| ====Bounds on Reliability====
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| The upper and lower bounds on reliability are given by:
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| ::<math>\begin{align}
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| & {{R}_{U}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}} \\
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| & {{R}_{L}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}}
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| \end{align}</math>
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| :where:
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| ::<math>\begin{align}
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| & 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 })+ \\
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| & & +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 })+ \\
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| & & +2\left( \frac{\partial R}{\partial \lambda } \right)\left( \frac{\partial R}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
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| \end{align}</math>
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| ====Bounds on Time====
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| 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:
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| ::<math>\begin{align}
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| & {{u}_{u}}= & \ln {{T}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} \\
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| & {{u}_{L}}= & \ln {{T}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}
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| \end{align}</math>
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| Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> we get:
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{{{T}_{U}}}}\text{ (upper bound)} \\
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| & {{T}_{L}}= & {{e}^{{{T}_{L}}}}\text{ (lower bound)}
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| \end{align}</math>
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| The variance of <math>u</math> is estimated from:
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| ::<math>\begin{align}
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| & 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 })+ \\
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| & & +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 })+ \\
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| & & +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
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| \end{align}</math>
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| ====A Generalized Gamma Distribution Example====
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| The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [21].
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| ::<math>\begin{array}{*{35}{l}}
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| \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} \\
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| \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} \\
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| \text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\
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| \end{array}</math>
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| When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:
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| ::<math>\begin{align}
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| & \widehat{\mu }= & 4.23064 \\
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| & \widehat{\sigma }= & 0.509982 \\
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| & \widehat{\lambda }= & 0.307639
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| \end{align}</math>
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| 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:
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| ::<math>\begin{align}
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| & {{\lambda }_{u}}= & -0.592087 \\
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| & {{\lambda }_{u}}= & 1.20736
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| \end{align}</math>
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| 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.
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| In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next.
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| [[Image:ldagamma10.2.gif|thumb|center|500px| ]] | |
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| 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.
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