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==Approximate Confidence Bounds for the Eyring-Weibull==
#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]]
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===Bounds on the Parameters===
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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:
 
<br>
::<math>\begin{align}
  & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
& {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} 
\end{align}</math>
 
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:also:
 
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::<math>\begin{align}
  & {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \\
& {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})} 
\end{align}</math>
 
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:and:
 
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::<math>\begin{align}
  & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
& {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} 
\end{align}</math>
 
 
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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:
 
<br>
::<math>\left[ \begin{matrix}
  Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{A}) & Cov(\widehat{\beta },\widehat{B})  \\
  Cov(\widehat{A},\widehat{\beta }) & Var(\widehat{A}) & Cov(\widehat{A},\widehat{B})  \\
  Cov(\widehat{B},\widehat{\beta }) & Cov(\widehat{B},\widehat{A}) & Var(\widehat{B})  \\
\end{matrix} \right]={{\left[ \begin{matrix}
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}}  \\
\end{matrix} \right]}^{-1}}</math>
 
===Confidence Bounds on Reliability===
 
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The reliability function for the Eyring-Weibull model (ML estimate) is given by:
 
<br>
::<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:
 
<br>
::<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>
 
<br>
:Setting:
 
<br>
::<math>\widehat{u}=\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]</math>
 
<br>
:or:
 
<br>
::<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> :
 
<br>
::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
 
<br>
::<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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<br>
::<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>
 
<br>
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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where  <math>{{u}_{U}}</math>  and  <math>{{u}_{L}}</math>  are estimated using Eqns (EyrExpu) and (EyrExpl).
 
===Confidence Bounds on Time===
 
<br>
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:
 
<br>
::<math>\begin{align}
  & \ln (R)= & -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\
& \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right) 
\end{align}</math>
 
<br>
:or:
 
<br>
::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>
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where 
<br>
::<math></math>
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The upper and lower bounds on  <math>\widehat{u}</math>  are then estimated from:
 
<br>
::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
 
<br>
::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
 
<br>
:where:
 
<br>
::<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})= & \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}) 
\end{align}</math>
 
<br>
The upper and lower bounds on time are then found by:
 
<br>
::<math>\begin{align}
  & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
& {{T}_{L}}= & {{e}^{{{u}_{L}}}} 
\end{align}</math>
 
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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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Latest revision as of 01:06, 17 August 2012

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