Temperature-Humidity Relationship

Appendix 9A: T-H Confidence Bounds

Approximate Confidence Bounds for the T-H Exponential

Confidence Bounds on the Mean Life

The mean life for the T-H exponential distribution is given by Eqn. (Temp-Hum) by setting [math]\displaystyle{ m=L(V) }[/math] . The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life (ML estimate of the mean life) are estimated by:

[math]\displaystyle{ {{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} }[/math]

[math]\displaystyle{ {{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \begin{align} Var(\widehat{m})=\ & {{e}^{2\left( \tfrac{\widehat{\phi }}{V}+\tfrac{\widehat{b}}{U} \right)}}[Var(\widehat{A})+\frac{{{\widehat{A}}^{2}}}{{{V}^{2}}}Var(\widehat{\phi }) +\frac{{{\widehat{A}}^{2}}}{{{U}^{2}}}Var(\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} }[/math]

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

[math]\displaystyle{ \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} }[/math]

Confidence Bounds on Reliability

The bounds on reliability at a given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

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:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

Approximate Confidence Bounds for the T-H Weibull

Bounds on the Parameters

Using the same approach as previously discussed ( [math]\displaystyle{ \widehat{\beta } }[/math] and [math]\displaystyle{ \widehat{A} }[/math] positive parameters):

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \begin{align} & {{b}_{U}}= & \widehat{b}+{{K}_{\alpha }}\sqrt{Var(\widehat{b})} \\ & {{b}_{L}}= & \widehat{b}-{{K}_{\alpha }}\sqrt{Var(\widehat{b})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\phi }_{U}}= & \widehat{\phi }+{{K}_{\alpha }}\sqrt{Var(\widehat{\phi })} \\ & {{\phi }_{L}}= & \widehat{\phi }-{{K}_{\alpha }}\sqrt{Var(\widehat{\phi })} \end{align} }[/math]

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

[math]\displaystyle{ \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}} }[/math]

where:

[math]\displaystyle{ 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] }[/math]

Confidence Bounds on Reliability

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

[math]\displaystyle{ \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 }}}}} }[/math]

or:

[math]\displaystyle{ \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]}}}} }[/math]

Setting:

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

or:

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

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V,U)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ u }[/math] :

[math]\displaystyle{ {{\widehat{u}}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{\widehat{u}}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \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} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

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:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{A}+\frac{\widehat{\phi }}{V}+\frac{\widehat{b}}{U} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T}. }[/math]

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \begin{align} Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{\widehat{A}}^{2}}}Var(\widehat{A}) +\frac{1}{{{U}^{2}}}Var(\widehat{b})+\frac{1}{{{V}^{2}}}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} }[/math]

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

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

Approximate Confidence Bounds for the T-H Lognormal

Bounds on the Parameters

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and [math]\displaystyle{ \widehat{A} }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (\widehat{A}) }[/math] are treated as normally distributed and the bounds are estimated from:

[math]\displaystyle{ \begin{align} {{\sigma }_{U}}=\ & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}&\text{ (Upper bound)} \\ {{\sigma }_{L}}=\ & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}&\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \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}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{A})}}{\widehat{A}}}}}&\text{ (Lower bound)} \end{align} }[/math]

The lower and upper bounds on [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ b }[/math] are estimated from:

[math]\displaystyle{ \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} }[/math]

and:

[math]\displaystyle{ \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} }[/math]

The variances and covariances of [math]\displaystyle{ A }[/math] , [math]\displaystyle{ \phi , }[/math] [math]\displaystyle{ b, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{A} }[/math] , [math]\displaystyle{ \widehat{\phi }, }[/math] [math]\displaystyle{ \widehat{b} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left( \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\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 }}_{{{T}'}}},\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 }}_{{{T}'}}} \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}} }[/math]

where:

[math]\displaystyle{ {{F}^{-1}}={{\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial b} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\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 }_{{{T}'}}}} & -\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 }_{{{T}'}}}} & -\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}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is given by:

[math]\displaystyle{ R({T}',V,U;A,\phi ,b,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

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

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \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 }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +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 }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial \phi } \right)}_{\widehat{\phi }}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial b} \right)}_{\widehat{b}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{A}^{2}}}Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{\phi })+\frac{1}{{{U}^{2}}}Var(\widehat{b})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\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 }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{U}Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

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:

[math]\displaystyle{ {T}'(V,U;\widehat{A},\widehat{\phi },\widehat{b},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{A})+\frac{\widehat{\phi }}{V}+\frac{\widehat{b}}{U}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

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

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

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

[math]\displaystyle{ \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 }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +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 }_{{{T}'}}}} \right)Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial \phi } \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial b} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{A}^{2}}}Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{\phi }) \\ & & +\frac{1}{{{U}^{2}}}Var(\widehat{b})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\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 }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{U}Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \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} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]

Appendix 9A: T-H Confidence Bounds

Approximate Confidence Bounds for the T-H Exponential

Confidence Bounds on the Mean Life

The mean life for the T-H exponential distribution is given by Eqn. (Temp-Hum) by setting [math]\displaystyle{ m=L(V) }[/math] . The upper [math]\displaystyle{ ({{m}_{U}}) }[/math] and lower [math]\displaystyle{ ({{m}_{L}}) }[/math] bounds on the mean life (ML estimate of the mean life) are estimated by:

[math]\displaystyle{ {{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} }[/math]

[math]\displaystyle{ {{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \widehat{m} }[/math] is given by:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \begin{align} Var(\widehat{m})=\ & {{e}^{2\left( \tfrac{\widehat{\phi }}{V}+\tfrac{\widehat{b}}{U} \right)}}[Var(\widehat{A})+\frac{{{\widehat{A}}^{2}}}{{{V}^{2}}}Var(\widehat{\phi }) +\frac{{{\widehat{A}}^{2}}}{{{U}^{2}}}Var(\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} }[/math]

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

[math]\displaystyle{ \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} }[/math]

Confidence Bounds on Reliability

The bounds on reliability at a given time, [math]\displaystyle{ T }[/math] , are estimated by:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\ & {{R}_{L}}= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} \end{align} }[/math]

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:

[math]\displaystyle{ \widehat{T}=-\widehat{m}\cdot \ln (R) }[/math]

The corresponding confidence bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\ & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) \end{align} }[/math]

Approximate Confidence Bounds for the T-H Weibull

Bounds on the Parameters

Using the same approach as previously discussed ( [math]\displaystyle{ \widehat{\beta } }[/math] and [math]\displaystyle{ \widehat{A} }[/math] positive parameters):

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \begin{align} & {{b}_{U}}= & \widehat{b}+{{K}_{\alpha }}\sqrt{Var(\widehat{b})} \\ & {{b}_{L}}= & \widehat{b}-{{K}_{\alpha }}\sqrt{Var(\widehat{b})} \end{align} }[/math]

and:

[math]\displaystyle{ \begin{align} & {{\phi }_{U}}= & \widehat{\phi }+{{K}_{\alpha }}\sqrt{Var(\widehat{\phi })} \\ & {{\phi }_{L}}= & \widehat{\phi }-{{K}_{\alpha }}\sqrt{Var(\widehat{\phi })} \end{align} }[/math]

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

[math]\displaystyle{ \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}} }[/math]

where:

[math]\displaystyle{ 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] }[/math]

Confidence Bounds on Reliability

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

[math]\displaystyle{ \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 }}}}} }[/math]

or:

[math]\displaystyle{ \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]}}}} }[/math]

Setting:

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

or:

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

The reliability function now becomes:

[math]\displaystyle{ \widehat{R}(T,V,U)={{e}^{-{{e}^{\widehat{u}}}}} }[/math]

The next step is to find the upper and lower bounds on [math]\displaystyle{ u }[/math] :

[math]\displaystyle{ {{\widehat{u}}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{\widehat{u}}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \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} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\ & {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} \end{align} }[/math]

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:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{A}+\frac{\widehat{\phi }}{V}+\frac{\widehat{b}}{U} }[/math]

where [math]\displaystyle{ \widehat{u}=\ln \widehat{T}. }[/math]

The upper and lower bounds on [math]\displaystyle{ u }[/math] are estimated from:

[math]\displaystyle{ {{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

[math]\displaystyle{ {{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})} }[/math]

where:

[math]\displaystyle{ \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} }[/math]

or:

[math]\displaystyle{ \begin{align} Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{\widehat{A}}^{2}}}Var(\widehat{A}) +\frac{1}{{{U}^{2}}}Var(\widehat{b})+\frac{1}{{{V}^{2}}}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} }[/math]

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

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\ & {{T}_{L}}= & {{e}^{{{u}_{L}}}} \end{align} }[/math]

Approximate Confidence Bounds for the T-H Lognormal

Bounds on the Parameters

Since the standard deviation, [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}} }[/math] , and [math]\displaystyle{ \widehat{A} }[/math] are positive parameters, [math]\displaystyle{ \ln ({{\widehat{\sigma }}_{{{T}'}}}) }[/math] and [math]\displaystyle{ \ln (\widehat{A}) }[/math] are treated as normally distributed and the bounds are estimated from:

[math]\displaystyle{ \begin{align} {{\sigma }_{U}}=\ & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}&\text{ (Upper bound)} \\ {{\sigma }_{L}}=\ & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}&\text{ (Lower bound)} \end{align} }[/math]

and:

[math]\displaystyle{ \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}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{A})}}{\widehat{A}}}}}&\text{ (Lower bound)} \end{align} }[/math]

The lower and upper bounds on [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ b }[/math] are estimated from:

[math]\displaystyle{ \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} }[/math]

and:

[math]\displaystyle{ \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} }[/math]

The variances and covariances of [math]\displaystyle{ A }[/math] , [math]\displaystyle{ \phi , }[/math] [math]\displaystyle{ b, }[/math] and [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{A} }[/math] , [math]\displaystyle{ \widehat{\phi }, }[/math] [math]\displaystyle{ \widehat{b} }[/math] , [math]\displaystyle{ {{\widehat{\sigma }}_{{{T}'}}}), }[/math] as follows:

[math]\displaystyle{ \left( \begin{matrix} Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\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 }}_{{{T}'}}},\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 }}_{{{T}'}}} \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}} }[/math]

where:

[math]\displaystyle{ {{F}^{-1}}={{\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial b} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\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 }_{{{T}'}}}} & -\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 }_{{{T}'}}}} & -\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}} }[/math]

Bounds on Reliability

The reliability of the lognormal distribution is given by:

[math]\displaystyle{ R({T}',V,U;A,\phi ,b,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt }[/math]

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

[math]\displaystyle{ R(\widehat{z})=\mathop{}_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

The bounds on [math]\displaystyle{ z }[/math] are estimated from:

[math]\displaystyle{ \begin{align} & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\ & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \end{align} }[/math]

where:

[math]\displaystyle{ \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 }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +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 }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial \phi } \right)}_{\widehat{\phi }}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2{{\left( \frac{\partial \widehat{z}}{\partial b} \right)}_{\widehat{b}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{A}^{2}}}Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{\phi })+\frac{1}{{{U}^{2}}}Var(\widehat{b})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\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 }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{U}Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right)] \end{align} }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ \begin{align} & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\ & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} \end{align} }[/math]

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:

[math]\displaystyle{ {T}'(V,U;\widehat{A},\widehat{\phi },\widehat{b},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{A})+\frac{\widehat{\phi }}{V}+\frac{\widehat{b}}{U}+z\cdot {{\widehat{\sigma }}_{{{T}'}}} }[/math]

where:

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

and:

[math]\displaystyle{ \Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz }[/math]

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

[math]\displaystyle{ \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 }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +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 }_{{{T}'}}}} \right)Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial \phi } \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) \\ & & +2\left( \frac{\partial {T}'}{\partial b} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

or:

[math]\displaystyle{ \begin{align} & Var({T}')= & \frac{1}{{{A}^{2}}}Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{\phi }) \\ & & +\frac{1}{{{U}^{2}}}Var(\widehat{b})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\ & & +\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 }}_{{{T}'}}} \right) \\ & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{U}Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \end{align} }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \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} }[/math]

Solving for [math]\displaystyle{ {{T}_{U}} }[/math] and [math]\displaystyle{ {{T}_{L}} }[/math] yields:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\ & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)} \end{align} }[/math]