Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on .CE.BB

2012-08-27T00:31:02Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on .CE.BB

← Older revision		Revision as of 00:31, 27 August 2012
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	~~====Bounds on <math>\lambda </math>====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on__.CE.BB]]
	~~=====Fisher Matrix Bounds=====~~
	~~The parameter <math>\lambda </math> must be positive, thus <math>\ln \lambda </math> is approximately treated as being normally distributed. These bounds are based on:~~


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

	~~<br>~~
	~~The approximate confidence bounds on <math>\lambda </math> are given as:~~


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


	~~where <math>\widehat{\lambda }=\tfrac{n}{T_{K}^{{\hat{\beta }}}}</math>~~ .
	~~The variance calculation is the same as Eqns~~. ~~(var1), (var2) and (var3).~~
	~~<br>~~
	~~<br>~~
	~~=====Crow Bounds=====~~
	~~''Time Terminated''~~
	~~<br>~~
	~~The confidence bounds on <math>\lambda </math> for time terminated data are calculated using:~~


	~~::<math>\begin{align}~~
	~~& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\~~
	~~& {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}~~
	~~\end{align}</math>~~



	~~''Failure Terminated''~~
	~~<br>~~
	~~The confidence bounds on <math>\lambda </math> for failure terminated data are calculated using:~~


	~~::<math>\begin{align}~~
	~~& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\~~
	~~& {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}~~
	~~\end{align}</math>~~

Nicolette Young: Created page with '====Bounds on $\lambda$ ==== =====Fisher Matrix Bounds===== The parameter $\lambda$ must be positive, thus $\ln \lambda$ is approximately …'

2012-01-10T22:37:44Z

Created page with '====Bounds on <math>\lambda </math>==== =====Fisher Matrix Bounds===== The parameter <math>\lambda </math> must be positive, thus <math>\ln \lambda </math> is approximately …'

New page

====Bounds on <math>\lambda </math>====
=====Fisher Matrix Bounds=====
The parameter <math>\lambda </math> must be positive, thus <math>\ln \lambda </math> is approximately treated as being normally distributed. These bounds are based on:

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

 
The approximate confidence bounds on <math>\lambda </math> are given as:

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

where <math>\widehat{\lambda }=\tfrac{n}{T_{K}^{{\hat{\beta }}}}</math> .
The variance calculation is the same as Eqns. (var1), (var2) and (var3).
 
 
=====Crow Bounds=====
''Time Terminated''
 
The confidence bounds on <math>\lambda </math> for time terminated data are calculated using:

::<math>\begin{align}
& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\
& {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}
\end{align}</math>

''Failure Terminated''
 
The confidence bounds on <math>\lambda </math> for failure terminated data are calculated using:

::<math>\begin{align}
& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\
& {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}
\end{align}</math>

Template:Bounds on lambda rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on .CE.BB

Nicolette Young: Created page with '====Bounds on \lambda ==== =====Fisher Matrix Bounds===== The parameter \lambda must be positive, thus \ln \lambda is approximately …'

Nicolette Young: Created page with '====Bounds on $\lambda$ ==== =====Fisher Matrix Bounds===== The parameter $\lambda$ must be positive, thus $\ln \lambda$ is approximately …'