Template:Acb4weib on parameters: Difference between revisions

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====Bounds on the Parameters====
#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]]
<br>
From the asymptotically normal property of the maximum likelihood estimators, and since  <math>\widehat{\beta },</math>  and  <math>\widehat{C}</math>  are positive parameters,  <math>\ln (\widehat{\beta }),</math>  and  <math>\ln (\widehat{C})</math>  can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be 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:
 
<br>
::<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>
 
<br>
and:
 
<br>
::<math>\begin{align}
  & {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\
& {{C}_{L}}= \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} 
\end{align}</math>
 
<br>
The variances and covariances of  <math>\beta ,</math>  <math>B,</math>  and  <math>C</math>  are estimated from the local Fisher matrix (evaluated at  <math>\widehat{\beta },</math>  <math>\widehat{B},</math>  <math>\widehat{C})</math> , as follows:
 
<br>
::<math>\left[ \begin{matrix}
  Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C})  \\
  Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C})  \\
  Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C})  \\
\end{matrix} \right]={{\left[ \begin{matrix}
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}}  \\
\end{matrix} \right]}^{-1}}</math>

Latest revision as of 05:59, 16 August 2012

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