Temperature-Humidity Relationship

=Appendix 9A: T-H Confidence Bounds=

Confidence Bounds on the Mean Life
The mean life for the T-H exponential distribution is given by Eqn. (Temp-Hum) 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 A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial \phi } \right)}^{2}}Var(\widehat{\phi }) +{{\left( \frac{\partial m}{\partial b} \right)}^{2}}Var(\widehat{b}) +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial \phi } \right)Cov(\widehat{A},\widehat{\phi }) \\ & +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial b} \right)Cov(\widehat{A},\widehat{b}) +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial \phi } \right)Cov(\widehat{\phi },\widehat{b}) \end{align}$$ or:


 * $$\begin{align}

Var(\widehat{m})=\ & {{e}^{2\left( \tfrac{\widehat{\phi }}{V}+\tfrac{\widehat{b}}{U} \right)}}[Var(\widehat{A})+\fracVar(\widehat{\phi }) +\fracVar(\widehat{b}) \\ & +\frac{2\widehat{A}}{V}Cov(\widehat{A},\widehat{\phi })+\frac{2\widehat{A}}{U}Cov(\widehat{A},\widehat{b}) +\frac{2{{\widehat{A}}^{2}}}{V\cdot U}Cov(\widehat{\phi },\widehat{b})] \end{align}$$

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


 * $$\left[ \begin{matrix}

Var(\widehat{A}) & Cov(\widehat{A},\widehat{\phi }) & Cov(\widehat{A},\widehat{b}) \\ Cov(\widehat{\phi },\widehat{A}) & Var(\widehat{\phi }) & Cov(\widehat{\phi },\widehat{b}) \\ Cov(\widehat{b},\widehat{A}) & Cov(\widehat{b},\widehat{\phi }) & Var(\widehat{b}) \\ \end{matrix} \right]=\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial b} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\phi }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial b} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{b}^{2}}} \\ \end{matrix} \right]_{}^{-1}$$

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 (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time or:


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

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  $$\widehat{A}$$  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}

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


 * $$\begin{align}

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

and:


 * $$\begin{align}

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

The variances and covariances of $$\beta ,$$   $$A,$$   $$b,$$  and  $$\phi $$  are estimated from the local Fisher matrix (evaluated at  $$\widehat{\beta },$$   $$\widehat{A},$$   $$\widehat{b},$$   $$\widehat{\phi })$$  as follows:


 * $$\left[ \begin{matrix}

Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{A}) & Cov(\widehat{\beta },\widehat{b}) & Cov(\widehat{\beta },\widehat{\phi }) \\ Cov(\widehat{A},\widehat{\beta }) & Var(\widehat{A}) & Cov(\widehat{A},\widehat{b}) & Cov(\widehat{A},\widehat{\phi }) \\ Cov(\widehat{b},\widehat{\beta }) & Cov(\widehat{b},\widehat{A}) & Var(\widehat{b}) & Cov(\widehat{b},\widehat{\phi }) \\ Cov(\widehat{\phi },\widehat{\beta }) & Cov(\widehat{\phi },\widehat{A}) & Cov(\widehat{\phi },\widehat{b}) & Var(\widehat{\phi }) \\ \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 A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial b} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial \phi } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial b} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial \phi } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{b}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial \phi } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial b} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\phi }^{2}}} \\ \end{matrix} \right]$$

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


 * $$\widehat{R}(T,V,U)={{e}^{-{{\left( \tfrac{T}{\widehat{A}}{{e}^{-\left( \tfrac{\widehat{\phi }}{V}+\tfrac{\widehat{b}}{U} \right)}} \right)}^{\widehat{\beta }}}}}$$

or:


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

Setting:


 * $$\widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{A}}{{e}^{-\left( \tfrac{\widehat{\phi }}{V}+\tfrac{\widehat{b}}{U} \right)}} \right)}^{\widehat{\beta }}} \right]$$

or:


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

The reliability function now becomes:


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

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


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


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

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

or:


 * $$\begin{align}

Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{\widehat{A}} \right)}^{2}}Var(\widehat{A}) +{{\left( \frac{\widehat{\beta }}{U} \right)}^{2}}Var(\widehat{b})+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{\phi }) -\frac{2\widehat{u}}{\widehat{A}}Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{U}Cov(\widehat{\beta },\widehat{b})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{\phi }) \\ & +\frac{2{{\widehat{\beta }}^{2}}}{\widehat{A}U}Cov(\widehat{A},\widehat{b})+\frac{2{{\widehat{\beta }}^{2}}}{\widehat{A}V}Cov(\widehat{A},\widehat{\phi }) +\frac{2{{\widehat{\beta }}^{2}}}{UV}Cov(\widehat{\phi },\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}$$

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

or:


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

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

or:


 * $$\begin{align}

Var(\widehat{u})= & \frac{1}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}Var(\widehat{A}) +\frac{1}Var(\widehat{b})+\frac{1}Var(\widehat{\phi }) +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{A}}Cov(\widehat{\beta },\widehat{A}) -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}U}Cov(\widehat{\beta },\widehat{b}) \\ & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{\phi }) +\frac{2}{\widehat{A}U}Cov(\widehat{A},\widehat{b}) +\frac{2}{\widehat{A}V}Cov(\widehat{A},\widehat{\phi }) +\frac{2}{VU}Cov(\widehat{b},\widehat{\phi }) \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  $$\widehat{A}$$  are positive parameters,  $$\ln ({{\widehat{\sigma }}_})$$  and  $$\ln (\widehat{A})$$  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}

{{A}_{U}}=\ & \widehat{A}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{A})}}{\widehat{A}}}}&\text{ (Upper bound)} \\ {{A}_{L}}=\ & \frac{\widehat{A}}&\text{ (Lower bound)} \end{align}$$ The lower and upper bounds on $$\phi $$  and  $$b$$  are estimated from:


 * $$\begin{align}

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

and:


 * $$\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}$$

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


 * $$\left( \begin{matrix}

Var\left( {{\widehat{\sigma }}_} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{\phi },{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{b},{{\widehat{\sigma }}_} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{A} \right) & Var\left( \widehat{A} \right) & Cov\left( \widehat{A},\widehat{\phi } \right) & Cov\left( \widehat{A},\widehat{b} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{\phi } \right) & Cov\left( \widehat{\phi },\widehat{A} \right) & Var\left( \widehat{\phi } \right) & Cov\left( \widehat{\phi },\widehat{b} \right) \\ Cov\left( \widehat{b},{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{b},\widehat{A} \right) & Cov\left( \widehat{b},\widehat{\phi } \right) & Var\left( \widehat{b} \right) \\ \end{matrix} \right)={{F}^{-1}}$$

where:


 * $${{F}^{-1}}={{\left( \begin{matrix}

-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial b} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial b} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\phi }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \phi \partial b} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{b}^{2}}} \\ \end{matrix} \right)}^{-1}}$$

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


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

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


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

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

or:


 * $$\begin{align}

Var(\widehat{z})=\ & \frac{1}{\widehat{\sigma }_^{2}}[\frac{1}Var(\widehat{A})+\frac{1}Var(\widehat{\phi })+\frac{1}Var(\widehat{b})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_}) +\frac{2}{A\cdot V}Cov\left( \widehat{A},\widehat{\phi } \right)+\frac{2}{A\cdot U}Cov\left( \widehat{A},\widehat{b} \right) \\ & +\frac{2}{V\cdot U}Cov\left( \widehat{\phi },\widehat{b} \right)+\frac{2\widehat{z}}{A}Cov\left( \widehat{A},{{\widehat{\sigma }}_} \right) +\frac{2\widehat{z}}{V}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_} \right)+\frac{2\widehat{z}}{U}Cov\left( \widehat{b},{{\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:


 * $${T}'(V,U;\widehat{A},\widehat{\phi },\widehat{b},{{\widehat{\sigma }}_})=\ln (\widehat{A})+\frac{\widehat{\phi }}{V}+\frac{\widehat{b}}{U}+z\cdot {{\widehat{\sigma }}_}$$

where:


 * $$\begin{align}

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

and:


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

The next step is to calculate the variance of $${T}'(V,U;\widehat{A},\widehat{\phi },\widehat{b},{{\widehat{\sigma }}_})$$  as follows:


 * $$\begin{align}

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

or:


 * $$\begin{align}

Var({T}')=\ & \frac{1}Var(\widehat{A})+\frac{1}Var(\widehat{\phi }) +\frac{1}Var(\widehat{b})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_}) +\frac{2}{A\cdot V}Cov\left( \widehat{A},\widehat{\phi } \right)+\frac{2}{A\cdot U}Cov\left( \widehat{A},\widehat{b} \right) \\ & +\frac{2}{V\cdot U}Cov\left( \widehat{\phi },\widehat{b} \right)+\frac{2\widehat{z}}{A}Cov\left( \widehat{A},{{\widehat{\sigma }}_} \right) +\frac{2\widehat{z}}{V}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_} \right)+\frac{2\widehat{z}}{U}Cov\left( \widehat{b},{{\widehat{\sigma }}_} \right) \end{align}$$ 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}$$