|
|
| Line 1: |
Line 1: |
| ===Bounds on the Parameters===
| | #REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]] |
| <br>
| |
| | |
| 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>
| |
| | |
| <br>
| |
| also:
| |
| | |
| <br>
| |
| ::<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>
| |
| | |
| <br>
| |
| and:
| |
| | |
| <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>
| |
| 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>
| |
| | |
| <br>
| |