Template:Acb4weib on reliability: Difference between revisions

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====Confidence Bounds on Reliability====
#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]]
 
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The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:
 
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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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or:
 
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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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Setting:
 
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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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or:
 
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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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The reliability function now becomes:
 
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::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}}</math>
 
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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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::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
 
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where:
 
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::<math>\begin{align}
  & 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}) \\
&  & +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}) \\
&  & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) 
\end{align}</math>
 
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or:
 
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::<math>\begin{align}
  & 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}) \\
&  & -\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}) 
\end{align}</math>
 
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The upper and lower bounds on reliability are:
 
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::<math>\begin{align}
  & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
& {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} 
\end{align}</math>
 
 
where  <math>{{u}_{U}}</math>  and  <math>{{u}_{L}}</math>  are estimated from Eqns. (ArreibRupper) and (ArreibRlower).
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Latest revision as of 06:01, 16 August 2012

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