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| =Appendix 7A: Eyring Confidence Bounds=
| | #REDIRECT [[Eyring_Relationship#Eyring_Confidence_Bounds]] |
| <br>
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| ==Approximate Confidence Bounds for the Eyring-Exponential==
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| <br>
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| ===Confidence Bounds on Mean Life===
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| <br>
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| | |
| The mean life for the Eyring relationship l is given by Eqn. (eyring) by setting <math>m=L(V)</math> . The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life (ML estimate of the mean life) are estimated by:
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| <br>
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| ::<math>{{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>
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| | |
| <br>
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| ::<math>{{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>
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| <br>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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| <br>
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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| <br>
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| If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds. The variance of <math>\widehat{m}</math> is given by:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>Var(\widehat{m})=\frac{1}{{{V}^{2}}}{{e}^{-2\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}}\left[ Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B})-\frac{1}{V}Cov(\widehat{A},\widehat{B}) \right]</math>
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| <br>
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| The variances and covariance of <math>A</math> and <math>B</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{A}</math> , <math>\widehat{B})</math> as follows:
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| <br>
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| ::<math>\left[ \begin{matrix}
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| Var(\widehat{A}) & Cov(\widehat{A},\widehat{B}) \\
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| Cov(\widehat{B},\widehat{A}) & Var(\widehat{B}) \\
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| \end{matrix} \right]={{\left[ \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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| ===Confidence Bounds on Reliability===
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| The bounds on reliability at a given time, <math>T</math> , are estimated by:
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| <br>
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| ::<math>\begin{align}
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| & {{R}_{U}}= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\
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| & & \\
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| & {{R}_{L}}= & {{e}^{-\tfrac{T}{{{m}_{L}}}}}
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| \end{align}</math>
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| <br>
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| where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).
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| <br>
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| ===Confidence Bounds on Time===
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| <br>
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| The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:
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| <br>
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| ::<math>\widehat{T}=-\widehat{m}\cdot \ln (R)</math>
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| <br>
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| The corresponding confidence bounds are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\
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| & & \\
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| & {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R)
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| \end{align}</math>
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| <br>
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| where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).
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| | |
| ==Approximate Confidence Bounds for the Eyring-Weibull==
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| <br>
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| ===Bounds on the Parameters===
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| <br>
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| From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta }</math> is a positive parameter, <math>\ln (\widehat{\beta })</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
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| & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
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| \end{align}</math>
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| <br>
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| :also:
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| <br>
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| ::<math>\begin{align}
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| & {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \\
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| & {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})}
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| \end{align}</math>
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| <br>
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| :and:
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| <br>
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| ::<math>\begin{align}
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| & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
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| & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
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| \end{align}</math>
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| <br>
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| The variances and covariances of <math>\beta ,</math> <math>A,</math> and <math>B</math> are estimated from the Fisher matrix (evaluated at <math>\widehat{\beta },</math> <math>\widehat{A},</math> <math>\widehat{B})</math> as follows:
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| <br>
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| ::<math>\left[ \begin{matrix}
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| Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{A}) & Cov(\widehat{\beta },\widehat{B}) \\
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| Cov(\widehat{A},\widehat{\beta }) & Var(\widehat{A}) & Cov(\widehat{A},\widehat{B}) \\
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| Cov(\widehat{B},\widehat{\beta }) & Cov(\widehat{B},\widehat{A}) & Var(\widehat{B}) \\
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| \end{matrix} \right]={{\left[ \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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| | |
| ===Confidence Bounds on Reliability===
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| <br>
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| The reliability function for the Eyring-Weibull model (ML estimate) is given by:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-{{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}}}}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]}}}}</math>
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| <br>
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| :Setting:
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| <br>
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| ::<math>\widehat{u}=\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)+\ln (V)+\widehat{A}-\frac{\widehat{B}}{V} \right]</math>
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| <br>
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| The reliability function now becomes:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-e\widehat{^{u}}}}</math>
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| <br>
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| The next step is to find the upper and lower bounds on <math>\widehat{u}</math> :
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| <br>
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>\begin{align}
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| & 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}) \\
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| & & +{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial A} \right)Cov(\widehat{\beta },\widehat{A}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial A} \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) \\
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| & & +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) \\
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| & & -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on reliability are:
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| <br>
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| ::<math>\begin{align}
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| & {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
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| & {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
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| \end{align}</math>
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| <br>
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| where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated using Eqns (EyrExpu) and (EyrExpl).
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| | |
| ===Confidence Bounds on Time===
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| <br>
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| The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:
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| <br>
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| ::<math>\begin{align}
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| & \ln (R)= & -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\
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| & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right)
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>
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| <br>
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| where
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| <br>
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| ::<math></math>
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| <br>
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| The upper and lower bounds on <math>\widehat{u}</math> are then estimated from:
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| <br>
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>\begin{align}
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| & 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}) \\
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| & & +{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial A} \right)Cov(\widehat{\beta },\widehat{A}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial A} \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta }) \\
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| & & +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) \\
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| & & +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}}Cov(\widehat{\beta },\widehat{A})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B}) \\
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| & & -\frac{2}{V}Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on time are then found by:
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| <br>
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
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| & {{T}_{L}}= & {{e}^{{{u}_{L}}}}
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| \end{align}</math>
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| <br>
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| where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated using Eqns. (EyrTimeu) and (EyrTimel).
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| <br>
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| | |
| ==Approximate Confidence Bounds for the Eyring-Lognormal==
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| <br>
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| ===Bounds on the Parameters===
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| <br>
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| The lower and upper bounds on <math>A</math> and <math>B</math> are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})}\text{ (Upper bound)} \\
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| & {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| :and:
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| <br>
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| ::<math>\begin{align}
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| & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\
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| & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| Since the standard deviation, <math>{{\widehat{\sigma }}_{{T}',}}</math> is a positive parameter, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math> is treated as normally distributed, and the bounds are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
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| & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| The variances and covariances of <math>A,</math> <math>B,</math> and <math>{{\sigma }_{{{T}'}}}</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{A},</math> <math>\widehat{B}</math> , <math>{{\widehat{\sigma }}_{{{T}'}}})</math> as follows:
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| <br>
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| ::<math>\left( \begin{matrix}
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| Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{A} \right) & Var\left( \widehat{A} \right) & Cov\left( \widehat{A},\widehat{B} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Cov\left( \widehat{B},\widehat{A} \right) & Var\left( \widehat{B} \right) \\
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| \end{matrix} \right)={{[F]}^{-1}}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>F=\left( \begin{matrix}
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| -\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 B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
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| \end{matrix} \right)</math>
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| <br>
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| ===Bounds on Reliability===
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| <br>
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| The reliability of the lognormal distribution is given by:
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| <br>
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| ::<math>R({T}',V;A,B,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (V)+\widehat{A}-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt</math>
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| <br>
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| Let <math>\widehat{z}(t,V;A,B,{{\sigma }_{T}})=\tfrac{t+\ln (V)+\widehat{A}-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>
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| <br>
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| For <math>t={T}'</math> , <math>\widehat{z}=\tfrac{{T}'+\ln (V)+\widehat{A}-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for <math>t=\infty ,</math> <math>\widehat{z}=\infty .</math> The above equation then becomes:
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| <br>
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| ::<math>R(\widehat{z})=\mathop{}_{\widehat{z}({T}',V)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
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| <br>
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| The bounds on <math>z</math> are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\
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| & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})}
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| \end{align}</math>
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| | |
| <br>
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| :where:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial A} \right)_{\widehat{A}}^{2}Var(\widehat{A})+\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}}) \\
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| & & +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) \\
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| & & +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) \\
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| & & +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)
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| \end{align}</math>
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| | |
| <br>
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| :or:
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| | |
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
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| & & -\frac{2}{V}Cov\left( \widehat{A},\widehat{B} \right)-2\widehat{z}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)]
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| \end{align}</math>
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| | |
| <br>
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| The upper and lower bounds on reliability are:
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| | |
| <br>
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| ::<math>\begin{align}
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| & {{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)}
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| \end{align}</math>
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| | |
| ===Confidence Bounds on Time===
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| <br>
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| The bounds around time for a given lognormal percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:
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| <br>
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| ::<math>{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}})=-\ln (V)-\widehat{A}+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>\begin{align}
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| & {T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\
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| & z= & {{\Phi }^{-1}}\left[ F({T}') \right]
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| \end{align}</math>
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| <br>
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| :and:
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| <br>
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| ::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
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| <br>
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| The next step is to calculate the variance of <math>{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}}):</math>
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| <br>
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| ::<math>\begin{align}
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| & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
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| & & +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial B} \right)Cov\left( \widehat{A},\widehat{B} \right) \\
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| & & +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
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| \end{align}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>\begin{align}
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| & Var({T}')= & Var(\widehat{A})+\frac{1}{V}Var(\widehat{B})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
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| & & -\frac{2}{V}Cov\left( \widehat{A},\widehat{B} \right) \\
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| & & -2\widehat{z}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
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| \end{align}</math>
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| <br>
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| The upper and lower bounds are then found by:
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| <br>
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| ::<math>\begin{align}
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| & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\
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| & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}
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| \end{align}</math>
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| <br>
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| Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:
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| <br>
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\
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| & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
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| \end{align}</math>
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