Template:Acb4weib on time: Difference between revisions

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(Created page with '===Confidence Bounds on Time=== <br> The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time: <br> ::<math>\begin…')
 
 
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===Confidence Bounds on Time===
#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]]
<br>
The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:
 
<br>
::<math>\begin{align}
  & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\
& \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right) 
\end{align}</math>
 
<br>
or:
 
<br>
::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V}</math>
<br>
 
where  <math>\widehat{u}=\ln \widehat{T}</math> .
 
<br>
The upper and lower bounds on  <math>u</math>  are estimated from:
 
 
::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
 
<br>
::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
 
<br>
where:
 
<br>
::<math>\begin{align}
  & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\
&  & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\
&  & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C}) 
\end{align}</math>
 
<br>
or:
 
<br>
::<math>\begin{align}
  & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\
&  & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\
&  & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C}) 
\end{align}</math>
 
<br>
The upper and lower bounds on time can then found by:
 
<br>
::<math>\begin{align}
  & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
& {{T}_{L}}= & {{e}^{{{u}_{L}}}} 
\end{align}</math>
 
<br>
where  <math>{{u}_{U}}</math>  and  <math>{{u}_{L}}</math>  are estimated using Eqns. (ArreibTupper) and (ArreibTlower).

Latest revision as of 06:02, 16 August 2012

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