Template:Acb-w on the parameters: Difference between revisions

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====Bounds on the Parameters====
#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Lognormal]]
 
<br>
The lower and upper bounds on  <math>B</math>  are estimated from:
 
<br>
::<math>\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}</math>
 
<br>
Since the standard deviation,  <math>{{\widehat{\sigma }}_{{{T}'}}}</math> , and the parameter  <math>C</math>  are positive parameters,  <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math>  and  <math>\ln (C)</math>  are treated as normally distributed. The bounds are estimated from:
 
<br>
::<math>\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}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)} 
\end{align}</math>
 
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and:
 
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::<math>\begin{align}
  & {{\sigma }_{U}}= {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
& {{\sigma }_{L}}= \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)} 
\end{align}</math>
 
<br>
The variances and covariances of  <math>B,</math>  <math>C,</math>  and  <math>{{\sigma }_{{{T}'}}}</math>  are estimated from the local Fisher matrix (evaluated at  <math>\widehat{B},</math>  <math>\widehat{C}</math> ,  <math>{{\widehat{\sigma }}_{{{T}'}}}),</math>  as follows:
<br>
 
::<math>\left[ \begin{matrix}
  Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)  \\
  Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right)  \\
  Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\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 _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}}  \\
\end{matrix} \right]}^{-1}}</math>
<br>

Latest revision as of 06:05, 16 August 2012

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