Template:Eyring-log cb on reliability: Difference between revisions

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===Bounds on Reliability===
#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Lognormal]]
 
<br>
The reliability of the lognormal distribution is given by:
 
<br>
::<math>R({T}',V;A,B,{{\sigma }_{{{T}'}}})=\int_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (V)+\widehat{A}-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt</math>
 
<br>
Let  <math>\widehat{z}(t,V;A,B,{{\sigma }_{T}})=\tfrac{t+\ln (V)+\widehat{A}-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},</math>  then  <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>
<br>
For  <math>t={T}'</math> ,  <math>\widehat{z}=\tfrac{{T}'+\ln (V)+\widehat{A}-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for  <math>t=\infty ,</math>  <math>\widehat{z}=\infty .</math>  The above equation then becomes:
 
<br>
::<math>R(\widehat{z})=\int_{\widehat{z}({T}',V)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
 
<br>
The bounds on  <math>z</math>  are estimated from:
 
<br>
::<math>\begin{align}
  & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\
& {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})} 
\end{align}</math>
 
<br>
where:
 
<br>
::<math>\begin{align}
  Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial A} \right)_{\widehat{A}}^{2}Var(\widehat{A})+\left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) +2{{\left( \frac{\partial \widehat{z}}{\partial A} \right)}_{\widehat{A}}}{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}Cov\left( \widehat{A},\widehat{B} \right) \\
& +2{{\left( \frac{\partial \widehat{z}}{\partial A} \right)}_{\widehat{A}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{A},{{\widehat{\sigma }}_{T}} \right) +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) 
\end{align}</math>
 
<br>
or:
 
 
<br>
::<math>\begin{align}
  & Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) -\frac{2}{V}Cov\left( \widehat{A},\widehat{B} \right)-2\widehat{z}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)]
\end{align}</math>
 
<br>
The upper and lower bounds on reliability are:
 
<br>
::<math>\begin{align}
  & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\
& {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)} 
\end{align}</math>

Latest revision as of 01:15, 17 August 2012

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