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| ====Bounds on the Parameters====
| | #REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]] |
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| 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}
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| & {{\beta }_{U}}= \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
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| & {{\beta }_{L}}= \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
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| \end{align}</math>
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| also:
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| ::<math>\begin{align}
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| & {{B}_{U}}= \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
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| & {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
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| \end{align}</math>
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| and:
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| ::<math>\begin{align}
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| & {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\
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| & {{C}_{L}}= \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}
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| \end{align}</math>
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| 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:
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| ::<math>\left[ \begin{matrix}
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| Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\
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| Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
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| Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
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| \end{matrix} \right]={{\left[ \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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