Template:Eyring-weib bounds on parameters

Bounds on the Parameters
From the asymptotically normal property of the maximum likelihood estimators, and since $$\widehat{\beta }$$  is a positive parameter,  $$\ln (\widehat{\beta })$$  can then be treated as normally distributed. After performing this transformation, the bounds on the parameters are estimated from:


 * $$\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}$$

also:


 * $$\begin{align}

& {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \\ & {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \end{align}$$

and:


 * $$\begin{align}

& {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\ & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \end{align}$$

The variances and covariances of $$\beta ,$$   $$A,$$  and  $$B$$  are estimated from the Fisher matrix (evaluated at  $$\widehat{\beta },$$   $$\widehat{A},$$   $$\widehat{B})$$  as follows:


 * $$\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}}$$