Template:Acb-l conf bounds for weib2

Confidence Bounds on Time
The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:


 * $$\begin{align}

& \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\ & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) \end{align}$$

or:


 * $$\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V}$$

where $$\widehat{u}=\ln \widehat{T}$$.

The upper and lower bounds on $$u$$  are estimated from:


 * $${{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}$$


 * $${{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}$$

where:


 * $$\begin{align}

& Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) \end{align}$$

or:


 * $$\begin{align}

& Var(\widehat{u})= & \frac{1}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}Var(\widehat{B})+\frac{1}Var(\widehat{C}) \\ & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) \end{align}$$

The upper and lower bounds on time can then found by:


 * $$\begin{align}

& {{T}_{U}}= & {{e}^} \\ & {{T}_{L}}= & {{e}^} \end{align}$$

where $${{u}_{U}}$$  and  $${{u}_{L}}$$  are estimated using Eqns. (ArreibTupper) and (ArreibTlower).