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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}
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| & {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})}\text{ (Upper bound)} \\
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| & {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| and:
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| ::<math>\begin{align}
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| & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\
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| & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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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}
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| {{\sigma }_{U}} &=\ {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
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| {{\sigma }_{L}} &=\ \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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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}
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| Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{A} \right) & Var\left( \widehat{A} \right) & Cov\left( \widehat{A},\widehat{B} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Cov\left( \widehat{B},\widehat{A} \right) & Var\left( \widehat{B} \right) \\
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| \end{matrix} \right)={{[F]}^{-1}}</math>
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| <br>
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| where:
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| <br>
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| ::<math>F=\left( \begin{matrix}
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| -\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} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
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| \end{matrix} \right)</math>
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| <br>
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