Template:Gamma distribution confidence bounds

Confidence Bounds
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.

Bounds on the Parameters
The lower and upper bounds on the mean, $$\widehat{\mu }$$, are estimated from:


 * $$\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}$$

Since the standard deviation, $$\widehat{\sigma }$$, must be positive,  $$\ln (\widehat{\sigma })$$  is treated as normally distributed and the bounds are estimated from:


 * $$\begin{align}

& {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}}}\text{ (upper bound)} \\ & {{k}_{L}}= & \frac{\widehat{\sigma }}\text{ (lower bound)} \end{align}$$

where $${{K}_{\alpha }}$$  is defined by:


 * $$\alpha =\frac{1}{\sqrt{2\pi }}\int_^{\infty }{{e}^{-\tfrac{2}}}dt=1-\Phi ({{K}_{\alpha }})$$

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

The variances and covariances of $$\widehat{\mu }$$  and  $$\widehat{k}$$  are estimated from the Fisher matrix, as follows:


 * $$\left( \begin{matrix}

\widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) \\ \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) & \widehat{Var}\left( \widehat{k} \right) \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} \\ {} & {} \\   -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}}  \\ \end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1}$$

$$\Lambda $$ is the log-likelihood function of the gamma distribution, described in chapter Parameter Estimation and Appendix: Distribution Log-Likelihood Equations

Bounds on Reliability
The reliability of the gamma distribution is:


 * $$\widehat{R}(t;\hat{\mu },\hat{k})=1-{{\Gamma }_{I}}(\widehat{k};{{e}^{\widehat{z}}})$$

where:


 * $$\widehat{z}=\ln (t)-\widehat{\mu }$$

The upper and lower bounds on reliability are:


 * $${{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (upper bound)}$$


 * $${{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (lower bound)}$$

where:


 * $$Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})$$

Bounds on Time
The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:


 * $$\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z$$

where:


 * $$z=\ln (-\ln (R))$$


 * $$Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })$$

or:


 * $$Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })$$

The upper and lower bounds are then found by:


 * $$\begin{align}

& {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\ & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)} \end{align}$$