Template:Acb4weib on parameters

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since [math]\displaystyle{ \widehat{\beta }, }[/math] and [math]\displaystyle{ \widehat{C} }[/math] are positive parameters, [math]\displaystyle{ \ln (\widehat{\beta }), }[/math] and [math]\displaystyle{ \ln (\widehat{C}) }[/math] can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

[math]\displaystyle{ \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]

also:

[math]\displaystyle{ \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]

and:

[math]\displaystyle{ \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]

The variances and covariances of [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] are estimated from the local Fisher matrix (evaluated at [math]\displaystyle{ \widehat{\beta }, }[/math] [math]\displaystyle{ \widehat{B}, }[/math] [math]\displaystyle{ \widehat{C}) }[/math] , as follows:

[math]\displaystyle{ \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]