Template:Cb for the eyring-weib

Confidence Bounds on Reliability
The reliability function for the Eyring-Weibull model (ML estimate) is given by:


 * $$\widehat{R}(T,V)={{e}^{-{{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}}}}$$


 * or:


 * $$\widehat{R}(T,V)={{e}^{-{{e}^{\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]}}}}$$


 * Setting:


 * $$\widehat{u}=\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]$$


 * or:


 * $$\widehat{u}=\widehat{\beta }\left[ \ln (T)+\ln (V)+\widehat{A}-\frac{\widehat{B}}{V} \right]$$

The reliability function now becomes:


 * $$\widehat{R}(T,V)={{e}^{-e\widehat{^{u}}}}$$

The next step is to find the upper and lower bounds on $$\widehat{u}$$ :


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


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


 * where:


 * $$\begin{align}

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


 * or:


 * $$\begin{align}

& Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) \\ & & +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\ & & +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) \\ & & -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B}) \end{align}$$

The upper and lower bounds on reliability are:


 * $$\begin{align}

& {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align}$$

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

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


 * $$\begin{align}

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


 * or:


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

where

The upper and lower bounds on $$\widehat{u}$$  are then 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 A} \right)}^{2}}Var(\widehat{A}) \\ & & +{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial A} \right)Cov(\widehat{\beta },\widehat{A}) \\ & & +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 A} \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{A},\widehat{B}) \end{align}$$


 * or:


 * $$\begin{align}

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

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


 * $$\begin{align}

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

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