Template:Eyring-weib bounds on parameters: Difference between revisions

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===Bounds on the Parameters===
#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]]
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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:
 
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::<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:
 
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::<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>
 
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Latest revision as of 01:07, 17 August 2012

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