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| ===Bounds on the Parameters===
| | #REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Lognormal]] |
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| <br>
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| The lower and upper bounds on <math>B</math> are estimated from:
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| <br>
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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> , 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:
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| <br>
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| ::<math>\begin{align}
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| & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\
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| & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| and:
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| <br>
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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>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:
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| <br>
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| ::<math>\left[ \begin{matrix}
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| Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\
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| \end{matrix} \right]=</math>
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| <br>
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| ::<math>={{\left[ \begin{matrix}
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| -\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} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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| <br>
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