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| ===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]].
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| ====Bounds on the Parameters====
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| The lower and upper bounds on the mean, <math>\widehat{\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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| 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:
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| ::<math>\begin{align}
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| & {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{{\hat{k}}}}}\text{ (upper bound)} \\
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| & {{k}_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{\widehat{k}}}}}\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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| 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{k}</math> are estimated from the Fisher matrix, as follows:
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| ::<math>\left( \begin{matrix}
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| \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) \\
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| \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) & \widehat{Var}\left( \widehat{k} \right) \\
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| \end{matrix} \right)=\left( \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} \\
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| {} & {} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}} \\
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| \end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1}</math>
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| <math>\Lambda </math> is the log-likelihood function of the gamma distribution, described in chapter [[Parameter Estimation]] and [[Appendix: Distribution Log-Likelihood Equations]]
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| ====Bounds on Reliability====
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| The reliability of the gamma distribution is:
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| ::<math>\widehat{R}(t;\hat{\mu },\hat{k})=1-{{\Gamma }_{I}}(\widehat{k};{{e}^{\widehat{z}}})</math>
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| where:
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| ::<math>\widehat{z}=\ln (t)-\widehat{\mu }</math>
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| The upper and lower bounds on reliability are:
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| ::<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>
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| ::<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>
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| where:
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| ::<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>
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| ====Bounds on Time====
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| The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:
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| ::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>
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| where:
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| ::<math>z=\ln (-\ln (R))</math>
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| ::<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>
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| or:
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| ::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>
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| The upper and lower bounds are then found by:
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| ::<math>\begin{align}
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| & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\
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| & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}
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| \end{align}</math>
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