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| ===Confidence Bounds on Mean Life===
| | #REDIRECT [[Eyring_Relationship#Eyring_Confidence_Bounds]] |
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| The mean life for the Eyring relationship l is given by Eqn. (eyring) by setting <math>m=L(V)</math> . The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life (ML estimate of the mean life) are estimated by:
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| ::<math>{{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>
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| ::<math>{{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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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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| If <math>\delta </math> is the confidence level, 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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| ::<math>\begin{align}
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| & Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
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| & & +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B})
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| \end{align}</math>
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| :or:
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| ::<math>Var(\widehat{m})=\frac{1}{{{V}^{2}}}{{e}^{-2\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}}\left[ Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B})-\frac{1}{V}Cov(\widehat{A},\widehat{B}) \right]</math>
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| The variances and covariance of <math>A</math> and <math>B</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{A}</math> , <math>\widehat{B})</math> as follows:
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| ::<math>\left[ \begin{matrix}
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| Var(\widehat{A}) & Cov(\widehat{A},\widehat{B}) \\
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| Cov(\widehat{B},\widehat{A}) & Var(\widehat{B}) \\
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| \end{matrix} \right]={{\left[ \begin{matrix}
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| -\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 A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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