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| ===Confidence Bounds on the Mean Life===
| | #REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Exponential]] |
| <br>
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| The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting <math>m=L(V)</math> as shown in Eqn. (Arrean). The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life are then estimated by:
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| <br>
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| ::<math>\begin{align}
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| & {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
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| & {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
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| \end{align}</math>
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| <br>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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| <br>
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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| <br>
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| If <math>\delta </math> is the confidence level (i.e., 95%=0.95), then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds. The variance of <math>\widehat{m}</math> is given by:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C})
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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>Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right]</math>
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| <br>
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| The variances and covariance of <math>B</math> and <math>C</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B}</math> , <math>\widehat{C})</math> as follows:
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| <br>
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| ::<math>\left[ \begin{matrix}
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| Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
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| Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
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| \end{matrix} \right]={{\left[ \begin{matrix}
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| -\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 B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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