Template:Bounds on cumulative mtbf camsaa-cb: Difference between revisions

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(Created page with '===Bounds on Cumulative MTBF=== ====Fisher Matrix Bounds==== The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is treated a…')
 
 
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===Bounds on Cumulative MTBF===
#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Cumulative_MTBF]]
====Fisher Matrix Bounds====
The cumulative MTBF,  <math>{{m}_{c}}(t)</math> , must be positive, thus  <math>\ln {{m}_{c}}(t)</math>  is treated as being normally distributed as well.
 
::<math>\frac{\ln {{{\hat{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \tilde{\ }\ N(0,1)</math>
 
The approximate confidence bounds on the cumulative MTBF are then estimated from:
 
::<math>CB={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}}</math>
<br>
:where:
<br>
::<math>{{\hat{m}}_{c}}(t)=\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}</math>
 
::<math>\begin{align}
  & Var({{{\hat{m}}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
&  & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\, 
\end{align}</math>
<br>
The variance calculation is the same as Eqn. (variance1) and:
 
::<math>\begin{align}
  & \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}\ln t \\
& \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}}{{t}^{1-\hat{\beta }}} 
\end{align}</math>
 
====Crow Bounds====
To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:
 
::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}</math>
 
::<math>C{{(t)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}</math>
 
:Then:
 
::<math>\begin{align}
  & {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\
& {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}} 
\end{align}</math>

Latest revision as of 04:06, 24 August 2012

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