Template:Eyring-log cb on parameters: Difference between revisions

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===Bounds on the Parameters===
#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Lognormal]]
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The lower and upper bounds on  <math>A</math>  and  <math>B</math>  are estimated from:
 
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::<math>\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}</math>
 
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and:
 
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::<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>
 
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Since the standard deviation,  <math>{{\widehat{\sigma }}_{{T}',}}</math>  is a positive parameter,  <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math>  is treated as normally distributed, and the bounds are estimated from:
 
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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>
 
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The variances and covariances of  <math>A,</math>  <math>B,</math>  and  <math>{{\sigma }_{{{T}'}}}</math>  are estimated from the local Fisher matrix (evaluated at  <math>\widehat{A},</math>  <math>\widehat{B}</math> ,  <math>{{\widehat{\sigma }}_{{{T}'}}})</math>  as follows:
 
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::<math>\left( \begin{matrix}
  Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)  \\
  Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{A} \right) & Var\left( \widehat{A} \right) & Cov\left( \widehat{A},\widehat{B} \right)  \\
  Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Cov\left( \widehat{B},\widehat{A} \right) & Var\left( \widehat{B} \right)  \\
\end{matrix} \right)={{[F]}^{-1}}</math>
 
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where:
 
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::<math>F=\left( \begin{matrix}
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}}  \\
\end{matrix} \right)</math>
 
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Latest revision as of 01:14, 17 August 2012

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