Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Instantaneous Failure Intensity

2012-08-27T00:34:39Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Instantaneous Failure Intensity

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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>~~

Nicolette Young: Created page with '====Bounds on Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== The instantaneous failure intensity, ${{\lambda }_{i}}(t)$ , must be positive, thus …'

2012-01-10T22:42:26Z

Created page with '====Bounds on Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== The instantaneous failure intensity, <math>{{\lambda }_{i}}(t)</math> , must be positive, thus …'

New page

====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>

Template:Bounds on instantaneous failure intensity rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Instantaneous Failure Intensity

Nicolette Young: Created page with '====Bounds on Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== The instantaneous failure intensity, {{\lambda }_{i}}(t) , must be positive, thus …'

Nicolette Young: Created page with '====Bounds on Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== The instantaneous failure intensity, ${{\lambda }_{i}}(t)$ , must be positive, thus …'