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

2012-08-27T00:37:24Z

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

← Older revision		Revision as of 00:37, 27 August 2012
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	~~====Bounds on Time Given Instantaneous Failure Intensity====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Time_Given_Instantaneous_Failure_Intensity]]
	~~=====Fisher Matrix Bounds=====~~
	~~These bounds are based on:~~

	~~::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left~~[ ~~\ln (\widehat{T}) \right~~]~~}}\sim 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>\begin{align}~~
	~~& 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 })~~
	~~\end{align}</math>~~

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

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


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


	~~=====Crow Bounds=====~~
	~~Step 1: Calculate <math>{{\lambda }_{i}}(T)=\tfrac{1}{MTB{{F}_{i}}}</math> .~~
	~~<br>~~
	~~Step 2: Use the equations from 13.1.7.9 to calculate the bounds on time given the instantaneous failure intensity.~~
	~~<br>~~
	~~<br>~~

Nicolette Young: Created page with '====Bounds on Time Given Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== These bounds are based on: :: $\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ …'$

2012-01-10T22:46:27Z

Created page with '====Bounds on Time Given Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== These bounds are based on: ::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ …'

New page

====Bounds on Time Given Instantaneous Failure Intensity====
=====Fisher Matrix Bounds=====
These bounds are based on:

::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\widehat{T}) \right]}}\sim 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>\begin{align}
& 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 })
\end{align}</math>

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

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

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

=====Crow Bounds=====
Step 1: Calculate <math>{{\lambda }_{i}}(T)=\tfrac{1}{MTB{{F}_{i}}}</math> .
<br>
Step 2: Use the equations from 13.1.7.9 to calculate the bounds on time given the instantaneous failure intensity.
<br>
<br>

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

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

Nicolette Young: Created page with '====Bounds on Time Given Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== These bounds are based on: ::\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ …'

Nicolette Young: Created page with '====Bounds on Time Given Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== These bounds are based on: :: $\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ …'$