Template:Generalized Gamma Confidence bounds

Confidence Bounds

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]\displaystyle{ \mu }[/math] are estimated from:

[math]\displaystyle{ \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]\displaystyle{ \widehat{\sigma } }[/math] , [math]\displaystyle{ \ln (\widehat{\sigma }) }[/math] is treated as normally distributed, and the bounds are estimated from:

[math]\displaystyle{ \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]\displaystyle{ \lambda , }[/math] the bounds are estimated from:

[math]\displaystyle{ \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]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds.

The variances and covariances of [math]\displaystyle{ \widehat{\mu } }[/math] and [math]\displaystyle{ \widehat{\sigma } }[/math] are estimated as follows:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ T }[/math] is a positive variable, the transformed variable [math]\displaystyle{ \hat{u}=\ln (\widehat{T}) }[/math] is treated as normally distributed and the bounds are estimated from:

[math]\displaystyle{ \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]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] we get:

[math]\displaystyle{ \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]\displaystyle{ u }[/math] is estimated from:

[math]\displaystyle{ \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]