Template:Bounds on time given instantaneous mtbf camsaa-cb: Difference between revisions

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(Created page with '===Bounds on Time Given Instantaneous MTBF=== ====Fisher Matrix Bounds==== The time, <math>T</math> , must be positive, thus <math>\ln T</math> is treated as being normally di…')
 
 
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===Bounds on Time Given Instantaneous MTBF===
#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Time_Given_Instantaneous_MTBF]]
====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>
 
<br>
Confidence bounds on the time are given by:
 
::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}</math>
 
<br>
: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>
 
<br>
The variance calculation is the same as Eqn. (variance1) and:
 
::<math>\hat{T}={{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}</math>
 
::<math>\begin{align}
  & \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot MTB{{F}_{i}} \right)}^{1/(1-\beta )}}\left[ \frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot MTB{{F}_{i}})+\frac{1}{\beta (1-\beta )} \right] \\
& \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}}{\lambda (1-\beta )} 
\end{align}</math>
 
====Crow Bounds====
:Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in Section 5.5.2.
:Step 2: Calculate the bounds on time as follows.
 
====Failure Terminated Data====
 
::<math>\hat{T}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{c})}^{1/(1-\beta )}}</math>
 
So the lower an upper bounds on time are:
 
::<math>{{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{1}}})}^{1/(1-\beta )}}</math>
 
::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})}^{1/(1-\beta )}}</math>
 
'''Time Terminated Data'''
 
::<math>\hat{T}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{\Pi })}^{1/(1-\beta )}}</math>
<br>
So the lower and upper bounds on time are:
 
::<math>{{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{1}}})}^{1/(1-\beta )}}</math>
 
::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}</math>

Latest revision as of 04:13, 24 August 2012

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