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| ==Approximate Confidence Bounds for the Arrhenius-Weibull:==
| | #REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]] |
| <br>
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| {{acb4weib on parameters}}
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| ===Confidence Bounds on Reliability===
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| <br>
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| The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \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)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \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( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \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 (\widehat{C})-\frac{\widehat{B}}{V} \right]</math>
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| <br>
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| The reliability function now becomes:
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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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| ::<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 B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\
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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 \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
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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})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\
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| & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
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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}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
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| & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
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| \end{align}</math>
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| where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated from Eqns. (ArreibRupper) and (ArreibRlower).
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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 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( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\
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| & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\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 \widehat{C}+\frac{\widehat{B}}{V}</math>
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| <br>
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| where <math>\widehat{u}=\ln \widehat{T}</math> .
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| <br>
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| The upper and lower bounds on <math>u</math> are estimated from:
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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 B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\
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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 \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
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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 })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\
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| & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\
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| & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on time can 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. (ArreibTupper) and (ArreibTlower).
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| {{appr conf bounds for arr-weibull}}
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