Template:Acb-w on the parameters

Bounds on the Parameters
The lower and upper bounds on $$B$$  are estimated from:


 * $$\begin{align}

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

Since the standard deviation, $${{\widehat{\sigma }}_}$$, and the parameter  $$C$$  are positive parameters,  $$\ln ({{\widehat{\sigma }}_})$$  and  $$\ln (C)$$  are treated as normally distributed. The bounds are estimated from:


 * $$\begin{align}

& {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\ & {{C}_{L}}= & \frac{\widehat{C}}\text{ (Lower bound)} \end{align}$$

and:


 * $$\begin{align}

& {{\sigma }_{U}}= & {{\widehat{\sigma }}_}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_})}}}}\text{ (Upper bound)} \\ & {{\sigma }_{L}}= & \frac\text{ (Lower bound)} \end{align}$$

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


 * $$\left[ \begin{matrix}

Var\left( {{\widehat{\sigma }}_} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\ \end{matrix} \right]=$$
 * $$={{\left[ \begin{matrix}

-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\ \end{matrix} \right]}^{-1}}$$