Template:App cb 4 the eyring-ex: Difference between revisions

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(Created page with '==Approximate Confidence Bounds for the Eyring-Exponential== <br> ===Confidence Bounds on Mean Life=== <br> The mean life for the Eyring relationship l is given by Eqn. (eyring)…')
 
 
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==Approximate Confidence Bounds for the Eyring-Exponential==
#REDIRECT [[Eyring_Relationship#Eyring_Confidence_Bounds]]
<br>
===Confidence Bounds on Mean Life===
<br>
 
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:
 
<br>
::<math>{{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>
 
<br>
::<math>{{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>
 
<br>
where  <math>{{K}_{\alpha }}</math>  is defined by:
 
<br>
::<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:
 
<br>
::<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:
 
<br>
::<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>
 
<br>
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:
 
<br>
::<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>
 
===Confidence Bounds on Reliability===
 
The bounds on reliability at a given time,  <math>T</math> , are estimated by:
 
<br>
::<math>\begin{align}
  & {{R}_{U}}= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\
&  &  \\
& {{R}_{L}}= & {{e}^{-\tfrac{T}{{{m}_{L}}}}} 
\end{align}</math>
 
<br>
where  <math>{{m}_{U}}</math>  and  <math>{{m}_{L}}</math>  are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).
<br>
===Confidence Bounds on Time===
<br>
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:
 
<br>
::<math>\widehat{T}=-\widehat{m}\cdot \ln (R)</math>
 
<br>
The corresponding confidence bounds are estimated from:
 
<br>
::<math>\begin{align}
  & {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\
&  &  \\
& {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R) 
\end{align}</math>
 
<br>
where  <math>{{m}_{U}}</math>  and  <math>{{m}_{L}}</math>  are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).

Latest revision as of 01:00, 17 August 2012

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