Template:Bounds on cumulative failure intensity rsa: 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>\…')
 
 
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====Bounds on Cumulative Failure Intensity====
#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#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 approximately treated as being normally distributed.
 
::<math>\frac{\ln ({{\widehat{\lambda }}_{c}}(t))-\ln ({{\lambda }_{c}}(t))}{\sqrt{Var\left[ \ln ({{\widehat{\lambda }}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)</math>
 
The approximate confidence bounds on the cumulative failure intensity are then estimated using:
 
::<math>CB={{\widehat{\lambda }}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\widehat{\lambda }}_{c}}(t))}/{{\widehat{\lambda }}_{c}}(t)}}</math>
 
:where:
 
::<math>{{\widehat{\lambda }}_{c}}(t)=\widehat{\lambda }{{t}^{\widehat{\beta }-1}}</math>
 
:and:
 
::<math>\begin{align}
  & Var({{\widehat{\lambda }}_{c}}(t))= & {{\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
&  & +2\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda }) 
\end{align}</math>
 
 
The variance calculation is the same as Eqns. (var1), (var2) and (var3):
 
::<math>\begin{align}
  & \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta }= & \widehat{\lambda }{{t}^{\widehat{\beta }-1}}\ln (t) \\
& \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda }= & {{t}^{\widehat{\beta }-1}} 
\end{align}</math>
 
<br>
=====Crow Bounds=====
The Crow cumulative failure intensity confidence bounds are given by:
 
::<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>

Latest revision as of 00:34, 27 August 2012

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