Template:Acb on the mean life: Difference between revisions

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====Confidence Bounds on the Mean Life====
#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Exponential]]
<br>
The Arrhenius-exponential distribution is given by setting  <math>m=L(V)</math>  in the exponential <math>pdf</math> equation. The upper  <math>({{m}_{U}})</math>  and lower  <math>({{m}_{L}})</math>  bounds on the mean life are then estimated by:
 
<br>
::<math>\begin{align}
& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} 
\end{align}</math>
 
<br>
where  <math>{{K}_{\alpha }}</math>  is defined by:
 
<br>
::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
 
<br>
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:
 
<br>
::<math>\begin{align}
  & 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}) +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) 
\end{align}</math>
 
<br>
or:
 
<br>
::<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>
 
<br>
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:
 
<br>
::<math>\left[ \begin{matrix}
  Var(\widehat{B}) & Cov(\widehat{B},\widehat{C})  \\
  Cov(\widehat{C},\widehat{B}) & Var(\widehat{C})  \\
\end{matrix} \right]={{\left[ \begin{matrix}
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C}  \\
  -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}}  \\
\end{matrix} \right]}^{-1}}</math>

Latest revision as of 05:48, 16 August 2012

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