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(Created page with '====Bounds on Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== The instantaneous failure intensity, <math>{{\lambda }_{i}}(t)</math> , must be positive, thus …')
 
 
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====Bounds on Instantaneous Failure Intensity====
#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Instantaneous_Failure_Intensity]]
=====Fisher Matrix Bounds=====
The instantaneous failure intensity,  <math>{{\lambda }_{i}}(t)</math> , must be positive, thus  <math>\ln {{\lambda }_{i}}(t)</math>  is approximately treated as being normally distributed.
 
::<math>\frac{\ln ({{\widehat{\lambda }}_{i}}(t))-\ln ({{\lambda }_{i}}(t))}{\sqrt{Var\left[ \ln ({{\widehat{\lambda }}_{i}}(t)) \right]}}\sim N(0,1)</math>
 
<br>
The approximate confidence bounds on the instantaneous failure intensity are then estimated from:
 
::<math>CB={{\widehat{\lambda }}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\widehat{\lambda }}_{i}}(t))}/{{\widehat{\lambda }}_{i}}(t)}}</math>
 
 
where  <math>{{\lambda }_{i}}(t)=\lambda \beta {{t}^{\beta -1}}</math>  and:
 
::<math>\begin{align}
  & Var({{\widehat{\lambda }}_{i}}(t))= & {{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
&  & +2\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda }) 
\end{align}</math>
 
<br>
The variance calculation is the same as Eqns. (var1), (var2) and (var3):
 
::<math>\begin{align}
  & \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\widehat{\beta }-1}}+\hat{\lambda }\hat{\beta }{{t}^{\widehat{\beta }-1}}\ln (t) \\
& \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda }= & \widehat{\beta }{{t}^{\widehat{\beta }-1}} 
\end{align}</math>
 
 
=====Crow Bounds=====
The Crow instantaneous failure intensity confidence bounds are given as:
 
::<math>\begin{align}
  & {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\
& {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}} 
\end{align}</math>

Latest revision as of 00:34, 27 August 2012

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