Template:Bounds on cumulative failure intensity camsaa-cb: Difference between revisions

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

Latest revision as of 04:08, 24 August 2012

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