Template:Bounds on time given cumulative failure intensity camsaa-gd: Difference between revisions

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(Created page with '===Bounds on Time Given Cumulative Failure Intensity=== ====Fisher Matrix Bounds==== The time, <math>T</math> , must be positive, thus <math>\ln T</math> is treated as being n…')
 
 
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===Bounds on Time Given Cumulative Failure Intensity===
#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Time_Given_Cumulative_Failure_Intensity_2]]
====Fisher Matrix Bounds====
The time,  <math>T</math> , must be positive, thus  <math>\ln T</math>  is treated as being normally distributed.
 
::<math>\frac{\ln \hat{T}-\ln T}{\sqrt{Var(\ln \hat{T}})}\ \tilde{\ }\ N(0,1)</math>
 
Confidence bounds on the time are given by:
 
::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}</math>
 
:where:
 
::<math>\begin{align}
  & Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
&  & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda }) 
\end{align}</math>
 
The variance calculation is the same as Eqn. (variances) and:
 
::<math>\begin{align}
  & \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\
& \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )} 
\end{align}</math>
 
 
====Crow Bounds====
:Step 1: Calculate:
 
::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}</math>
 
:Step 2: Estimate the number of failures:
 
::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}</math>
 
:Step 3: Obtain the confidence bounds on time given the cumulative failure intensity by solving for  <math>{{t}_{l}}</math>  and  <math>{{t}_{u}}</math>  in the following equations:
 
::<math>\begin{align}
  & {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\
& {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)} 
\end{align}</math>

Latest revision as of 03:52, 24 August 2012

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