Template:Eyring-log cb on parameters

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


 * $$\begin{align}

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


 * and:


 * $$\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, $${{\widehat{\sigma }}_{{T}',}}$$  is a positive parameter,  $$\ln ({{\widehat{\sigma }}_})$$  is treated as normally distributed, and the bounds are estimated from:


 * $$\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 $$A,$$   $$B,$$  and  $${{\sigma }_}$$  are estimated from the local Fisher matrix (evaluated at  $$\widehat{A},$$   $$\widehat{B}$$,  $${{\widehat{\sigma }}_})$$  as follows:


 * $$\left( \begin{matrix}

Var\left( {{\widehat{\sigma }}_} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{A} \right) & Var\left( \widehat{A} \right) & Cov\left( \widehat{A},\widehat{B} \right) \\ Cov\left( {{\widehat{\sigma }}_},\widehat{B} \right) & Cov\left( \widehat{B},\widehat{A} \right) & Var\left( \widehat{B} \right) \\ \end{matrix} \right)={{[F]}^{-1}}$$


 * where:


 * $$F=\left( \begin{matrix}

-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_}\partial B} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\ \end{matrix} \right)$$