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| ==Approximate Confidence Bounds for the Arrhenius-Lognormal==
| | #REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Lognormal]] |
| <br>
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| {{acb-w on the parameters}}
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| {{acb-w on reliability}}
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| ===Confidence Bounds on Time===
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| <br>
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| The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:
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| <br>
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| ::<math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>
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| <br>
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| where:
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| <br>
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| ::<math>\begin{align}
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| & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\
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| & z= & {{\Phi }^{-1}}\left[ F({T}') \right]
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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>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
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| <br>
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| The next step is to calculate the variance of <math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}):</math>
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| <br>
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| ::<math>\begin{align}
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| & Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
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| & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\
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| & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
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| \end{align}</math>
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| <br>
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| or:
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| <br>
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| ::<math>\begin{align}
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| & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
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| & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\
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| & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
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| \end{align}</math>
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| <br>
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| The upper and lower bounds are then found by:
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| <br>
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| ::<math>\begin{align}
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| & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\
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| & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}
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| \end{align}</math>
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| <br>
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| Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:
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| <br>
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\
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| & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
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| \end{align}</math>
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| {{RS Copyright}}
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