Template:Erying-weib cb on reliability: Difference between revisions

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===Confidence Bounds on Reliability===
#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]]
 
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The reliability function for the Eyring-Weibull model (ML estimate) is given by:
 
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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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or:
 
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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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Setting:
 
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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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or:
 
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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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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 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}) \\
& +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}) 
\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 })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\
& +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B}) 
\end{align}</math>
 
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The upper and lower bounds on reliability are:
 
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::<math>\begin{align}
  & {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
& {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}} 
\end{align}</math>
 
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Latest revision as of 01:08, 17 August 2012

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