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| ===Confidence Bounds on Time===
| | #REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]] |
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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 <math>\widehat{u}=ln(\widehat{T})</math>. 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}) +{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B}) +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial A} \right)Cov(\widehat{\beta },\widehat{A}) \\
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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}) +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 }) +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) +\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}) -\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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