Template:Eyring-weib cb on time: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
===Confidence Bounds on Time===
#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]]
 
<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>\begin{align}
  \ln (R)&=\  -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\
  \ln (-\ln (R))&=\  \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right) 
\end{align}</math>
 
<br>
or:
 
<br>
::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>
<br>
where 
<br>
::<math></math>
<br>
The upper and lower bounds on  <math>\widehat{u}</math>  are then estimated from:
 
<br>
::<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 A} \right)}^{2}}Var(\widehat{A}) +{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B}) +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial A} \right)Cov(\widehat{\beta },\widehat{A}) \\
& +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 A} \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{A},\widehat{B}) 
\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 }) +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}}Cov(\widehat{\beta },\widehat{A})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B}) -\frac{2}{V}Cov(\widehat{A},\widehat{B}) 
\end{align}</math>
 
<br>
The upper and lower bounds on time are then found by:
 
<br>
::<math>\begin{align}
  & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
& {{T}_{L}}= & {{e}^{{{u}_{L}}}} 
\end{align}</math>
 
<br>

Latest revision as of 01:10, 17 August 2012

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Eyring-weib_cb_on_time&oldid=32434"