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

2012-08-27T00:36:46Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Time Given Cumulative Failure Intensity

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	~~====Bounds on Time Given Cumulative Failure Intensity====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Time_Given_Cumulative_Failure_Intensity]]
	~~=====Fisher Matrix Bounds=====~~
	~~The time, <math>T</math> , must be positive, thus <math>\ln T</math> is approximately treated as being normally distributed.~~

	~~::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left~~[ ~~\ln \widehat{T} \right~~]~~}}\ \tilde{\ }\ N(0,1)</math>~~

	~~The confidence bounds on the time are given by:~~

	~~::<math>CB=\widehat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{T})}/\widehat{T}}}</math>~~

	~~:where:~~

	::<math>Var(\widehat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })</math>

	~~The variance calculation is the same as Eqns. (var1), (var2) and (var3):~~

	~~::<math>\widehat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}</math>~~


	~~::<math>\begin{align}~~
	~~& \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\~~
	~~& \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )}~~
	~~\end{align}</math>~~


	~~=====Crow Bounds=====~~
	~~Step 1: Calculate:~~


	~~::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}</math>~~


	~~Step 2: Estimate the number of failures:~~


	~~::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}</math>~~


	~~Step 3: Obtain the confidence bounds on time given the cumulative failure intensity by solving for <math>{{t}_{l}}</math> and <math>{{t}_{u}}</math> in the following equations:~~

	~~::<math>\begin{align}~~
	~~& {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\~~
	~~& {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)}~~
	~~\end{align}</math>~~

Nicolette Young: Created page with '====Bounds on Time Given Cumulative Failure Intensity==== =====Fisher Matrix Bounds===== The time, $T$ , must be positive, thus $\ln T$ is approximately …'

2012-01-10T22:44:51Z

Created page with '====Bounds on Time Given Cumulative Failure Intensity==== =====Fisher Matrix Bounds===== The time, <math>T</math> , must be positive, thus <math>\ln T</math> is approximately …'

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====Bounds on Time Given Cumulative Failure Intensity====
=====Fisher Matrix Bounds=====
The time, <math>T</math> , must be positive, thus <math>\ln T</math> is approximately treated as being normally distributed.

::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ \ln \widehat{T} \right]}}\ \tilde{\ }\ N(0,1)</math>

The confidence bounds on the time are given by:

::<math>CB=\widehat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{T})}/\widehat{T}}}</math>

:where:

::<math>Var(\widehat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })</math>

The variance calculation is the same as Eqns. (var1), (var2) and (var3):

::<math>\widehat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}</math>

::<math>\begin{align}
& \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\
& \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )}
\end{align}</math>

=====Crow Bounds=====
Step 1: Calculate:

::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}</math>

Step 2: Estimate the number of failures:

::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}</math>

Step 3: Obtain the confidence bounds on time given the cumulative failure intensity by solving for <math>{{t}_{l}}</math> and <math>{{t}_{u}}</math> in the following equations:

::<math>\begin{align}
& {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\
& {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)}
\end{align}</math>

Template:Bounds on time given cumulative failure intensity rsa - Revision history

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

Nicolette Young: Created page with '====Bounds on Time Given Cumulative Failure Intensity==== =====Fisher Matrix Bounds===== The time, T , must be positive, thus \ln T is approximately …'

Nicolette Young: Created page with '====Bounds on Time Given Cumulative Failure Intensity==== =====Fisher Matrix Bounds===== The time, $T$ , must be positive, thus $\ln T$ is approximately …'