Temperature-NonThermal Relationship

=Appendix 10A: T-NT Confidence Bounds=

Confidence Bounds on the Mean Life
The mean life for the T-NT model is given by setting $$m=L(V)$$. The upper $$({{m}_{U}})$$  and lower  $$({{m}_{L}})$$  bounds on the mean life (ML estimate of the mean life) are estimated by:


 * $${{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}$$


 * $${{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}$$

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 variance of $$\widehat{m}$$  is given by:


 * $$\begin{align}

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

or:


 * $$\begin{align}

Var(\widehat{m})= & \frac{1}{{e}^{2\tfrac{\widehat{B}}{V}}}[\fracVar(\widehat{B})+Var(\widehat{C}) +{{\widehat{C}}^{2}}{{\left( \ln (U) \right)}^{2}}Var(\widehat{n}) +\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \\ & -\frac{2{{\widehat{C}}^{2}}\ln (U)}{V}Cov(\widehat{B},\widehat{n}) -2\widehat{C}\ln (U)Cov(\widehat{C},\widehat{n})] \end{align}$$

The variances and covariance of $$B,$$   $$C$$  and  $$n$$  are estimated from the local Fisher matrix (evaluated at  $$\widehat{B},$$   $$\widehat{C},$$   $$\widehat{n})$$  as follows:


 * $$\left[ \begin{matrix}

Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) & Cov(\widehat{B},\widehat{n}) \\ Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) & Cov(\widehat{C},\widehat{n}) \\ Cov(\widehat{n},\widehat{B}) & Cov(\widehat{n},\widehat{C}) & Var(\widehat{n}) \\ \end{matrix} \right]={{\left[ F \right]}^{-1}}$$


 * where,


 * $$F=\left[ \begin{matrix}

-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{n}^{2}}} \\ \end{matrix} \right].$$

Confidence Bounds on Reliability
The bounds on reliability at a given time, $$T$$, are estimated by:


 * $$\begin{align}

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

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


 * $$\widehat{T}=-\widehat{m}\cdot \ln (R)$$

The corresponding confidence bounds are estimated from:


 * $$\begin{align}

& {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align}$$

Bounds on the Parameters
Using the same approach as previously discussed ( $$\widehat{\beta }$$ and $$\widehat{C}$$  positive parameters):


 * $$\begin{align}

& {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\ & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \end{align}$$


 * $$\begin{align}

& {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \end{align}$$


 * $$\begin{align}

& {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\ & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \end{align}$$

and:


 * $$\begin{align}

& {{n}_{U}}= & \widehat{n}+{{K}_{\alpha }}\sqrt{Var(\widehat{n})} \\ & {{n}_{L}}= & \widehat{n}-{{K}_{\alpha }}\sqrt{Var(\widehat{n})} \end{align}$$

The variances and covariances of $$\beta ,$$   $$B,$$   $$C,$$  and  $$n$$  are estimated from the Fisher matrix (evaluated at  $$\widehat{\beta },$$   $$\widehat{B},$$   $$\widehat{C},$$   $$\widehat{n})$$  as follows:


 * $$\left[ \begin{matrix}

Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) & Cov(\widehat{\beta },\widehat{n}) \\ Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) & Cov(\widehat{B},\widehat{n}) \\ Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) & Cov(\widehat{C},\widehat{n}) \\ Cov(\widehat{n},\widehat{\beta }) & Cov(\widehat{n},\widehat{B}) & Cov(\widehat{n},\widehat{C}) & Var(\widehat{n}) \\ \end{matrix} \right]={{\left[ F \right]}^{-1}}$$

where:


 * $$F=\left[ \begin{matrix}

-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{n}^{2}}} \\ \end{matrix} \right]$$

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


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

or:


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

Setting:


 * $$\widehat{u}=\ln \left[ {{\left( \frac{{{U}^{\widehat{n}}}{{e}^{-\tfrac{\widehat{B}}{V}}}}{\widehat{C}}T \right)}^{\widehat{\beta }}} \right]$$

or:


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

The reliability function now becomes:


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

The next step is to find the upper and lower bounds on $$u$$ :


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


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

where:


 * $$\begin{align}

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

or:


 * $$\begin{align}

Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) +{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C})+{{\left( \widehat{\beta }\ln (U) \right)}^{2}}Var(\widehat{n}) -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\ & +2\widehat{u}\ln (U)Cov(\widehat{\beta },\widehat{n}) +\frac{2{{\widehat{\beta }}^{2}}}{\widehat{C}V}Cov(\widehat{B},\widehat{C})-\frac{2{{\widehat{\beta }}^{2}}\ln (U)}{V}Cov(\widehat{B},\widehat{n}) -\frac{2{{\widehat{\beta }}^{2}}\ln (U)}{\widehat{C}}Cov(\widehat{C},\widehat{n}) \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}$$

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 as follows:


 * $$\begin{align}

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

or:


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

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

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


 * $$\begin{align}

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

Bounds on the Parameters
Since the standard deviation, $${{\widehat{\sigma }}_}$$, and  $$\widehat{C}$$  are positive parameters,  $$\ln ({{\widehat{\sigma }}_})$$  and  $$\ln (\widehat{C})$$  are treated as normally distributed and the bounds are estimated from:


 * $$\begin{align}

{{\sigma }_{U}}=\ & {{\widehat{\sigma }}_}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_})}}}} &\text{ (Upper bound)} \\ & &  \\  {{\sigma }_{L}}=\ & \frac &\text{ (Lower bound)} \end{align}$$ and:


 * $$\begin{align}

{{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} &\text{ (Upper bound)} \\ & &  \\  {{C}_{L}}= & \frac{\widehat{A}} &\text{ (Lower bound)} \end{align}$$

The lower and upper bounds on $$B$$  and  $$n$$  are estimated from:


 * $$\begin{align}

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

and:


 * $$\begin{align}

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

The variances and covariances of $$B$$,  $$C,$$   $$n,$$  and  $${{\sigma }_}$$  are estimated from the local Fisher matrix (evaluated at  $$\widehat{B},$$   $$\widehat{C},$$   $$\widehat{n}$$ ,  $${{\widehat{\sigma }}_})$$  as follows:


 * $$\left( \begin{matrix}

Var\left( {{\widehat{\sigma }}_} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) & Cov\left( \widehat{B},\widehat{n} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) & Cov\left( \widehat{C},\widehat{n} \right) \\ Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{n},\widehat{B} \right) & Cov\left( \widehat{n},\widehat{C} \right) & Var\left( \widehat{n} \right) \\ \end{matrix} \right)={{\left[ F \right]}^{-1}}$$


 * where:


 * $$F=\left( \begin{matrix}

-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial n} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial n\partial C} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{n}^{2}}} \\ \end{matrix} \right)$$

Bounds on Reliability
The reliability of the lognormal distribution is given by:


 * $$R({T}',U,V;B,C,n,{{\sigma }_})=\int_^{\infty }\frac{1}{{{\widehat{\sigma }}_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})+\widehat{n}\ln ({{U}_{i}})-\tfrac{\widehat{B}}{{{V}_{i}}}} \right)}^{2}}}}dt$$

Let $$\widehat{z}(t,U,V;B,C,n,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})+\widehat{n}\ln (U)-\tfrac{\widehat{B}}{V}},$$  then  $$\tfrac{d\widehat{z}}{dt}=\tfrac{1}.$$ For $$t={T}'$$,  $$\widehat{z}=\tfrac{{T}'-\ln (\widehat{C})+\widehat{n}\ln (U)-\tfrac{\widehat{B}}{V}}$$ , and for  $$t=\infty ,$$   $$\widehat{z}=\infty .$$ The above equation then becomes:


 * $$R(\widehat{z})=\int_{\widehat{z}({T}',U,V)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The bounds on $$z$$  are estimated from:


 * $$\begin{align}

& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align}$$

where:


 * $$\begin{align}

Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C}) +\left( \frac{\partial \widehat{z}}{\partial n} \right)_{\widehat{b}}^{2}Var(\widehat{n})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_}} \right)_^{2}Var({{\widehat{\sigma }}_}) +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\ & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial b} \right)}_{\widehat{n}}}Cov\left( \widehat{B},\widehat{n} \right) +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial n} \right)}_{\widehat{n}}}Cov\left( \widehat{C},\widehat{n} \right) +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_}} \right)}_}Cov\left( \widehat{B},{{\widehat{\sigma }}_} \right) \\ & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_}} \right)}_}Cov\left( \widehat{C},{{\widehat{\sigma }}_} \right) +2{{\left( \frac{\partial \widehat{z}}{\partial n} \right)}_{\widehat{n}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_}} \right)}_}Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right) \end{align}$$

or:


 * $$\begin{align}

Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_^{2}}[\frac{1}Var(\widehat{B})+\frac{1}Var(\widehat{C})+\ln {{(U)}^{2}}Var(\widehat{n})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_}) +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)-\frac{2\ln (U)}{V}Cov\left( \widehat{B},\widehat{n} \right) \\ & -\frac{2\ln (U)}{C}Cov\left( \widehat{C},\widehat{n} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_} \right) +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_} \right)-2\widehat{z}\ln (U)Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right)] \end{align}$$

The upper and lower bounds on reliability are:


 * $$\begin{align}

& {{R}_{U}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \int_^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align}$$

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


 * $${T}'(U,V;\widehat{B},\widehat{C},\widehat{n},{{\widehat{\sigma }}_})=\ln (\widehat{C})+\widehat{n}\ln (U)-\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_}$$

where:


 * $$\begin{align}

{T}'(U,V;\widehat{A},\widehat{\phi },\widehat{b},{{\widehat{\sigma }}_})=\ & \ln (T) \\ z=\ & {{\Phi }^{-1}}\left[ F({T}') \right] \end{align}$$

and:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',U,V)}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz$$

The next step is to calculate the variance of $${T}'(U,V;\widehat{B},\widehat{C},\widehat{n},{{\widehat{\sigma }}_})$$ :


 * $$\begin{align}

Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C}) +{{\left( \frac{\partial {T}'}{\partial n} \right)}^{2}}Var(\widehat{n})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)}^{2}}Var({{\widehat{\sigma }}_}) +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\ & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial n} \right)Cov\left( \widehat{B},\widehat{n} \right) +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial n} \right)Cov\left( \widehat{C},\widehat{n} \right) +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_} \right) \\ & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_} \right) +2\left( \frac{\partial {T}'}{\partial n} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_}} \right)Cov\left( \widehat{n},{{\widehat{\sigma }}_} \right) \end{align}$$

or:

The upper and lower bounds are then found by:


 * $$\begin{align}

& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\ & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')} \end{align}$$

Solving for $${{T}_{U}}$$  and  $${{T}_{L}}$$  yields:


 * $$\begin{align}

& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align}$$