Template:Eyring-ex cb on mean life: Difference between revisions

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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 is given 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 }}\int_{{{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}
  & 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}) +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B}) 
\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}
  Var(\widehat{A}) & Cov(\widehat{A},\widehat{B})  \\
  Cov(\widehat{B},\widehat{A}) & Var(\widehat{B})  \\
\end{matrix} \right]={{\left[ \begin{matrix}
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}}  \\
\end{matrix} \right]}^{-1}}</math>
 
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Latest revision as of 01:01, 17 August 2012

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