Template:Bounds on beta rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on .CE.B2

2012-08-27T00:30:19Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on .CE.B2

← Older revision		Revision as of 00:30, 27 August 2012
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	~~====Bounds on <math>\beta </math>====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on__.CE.B2]]
	~~=====Fisher Matrix Bounds=====~~
	~~The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is approximately treated as being normally distributed.~~


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


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


	~~::<math>\widehat{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\widehat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left~~[ (T_{q}^{\widehat{\beta }}\ln ({{T}_{q}})-S_{q}^{\widehat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln ({{X}_{i}}{{}_{q}})}</math>


	~~All variance can be calculated using the Fisher Information Matrix~~.
	~~<br>~~
	~~<math>\Lambda </math> is the natural log-likelihood function~~.


	::<math>\Lambda =\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ {{N}_{q}}(\ln (\lambda )+\ln (\beta ))-\lambda (T_{q}^{\beta }-S_{q}^{\beta })+(\beta -1)\underset{i=1}{\overset{{{N}_{q}}}{\mathop \sum }}\,\ln ({{x}_{iq}}) \right]~~</math>~~


	~~::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\lambda }^{2}}}</math>~~


	::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }\ln ({{T}_{q}})-S_{q}^{\beta }\ln ({{S}_{q}}) \right]</math>


	::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\beta }^{2}}}-\lambda \underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }{{(\ln ({{T}_{q}}))}^{2}}-S_{q}^{\beta }{{(\ln ({{S}_{q}}))}^{2}} \right]~~</math>~~

	~~=====Crow Bounds=====~~
	~~Calculate the conditional maximum likelihood estimate of <math>\tilde{\beta }</math> :~~


	::<math>\tilde{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{iq}}} \right)}</math>


	~~The Crow 2-sided <math>(1-a)</math> 100-percent confidence bounds on <math>\beta </math> are:~~

	~~::<math>\begin{align}~~
	~~& {{\beta }_{L}}= & \tilde{\beta }\frac{\chi _{\tfrac{\alpha }{2},2M}^{2}}{2M} \\~~
	~~& {{\beta }_{U}}= & \tilde{\beta }\frac{\chi _{1-\tfrac{\alpha }{2},2M}^{2}}{2M}~~
	~~\end{align}</math>~~

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

2012-01-10T22:31:58Z

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

New page

====Bounds on <math>\beta </math>====
=====Fisher Matrix Bounds=====
The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is approximately treated as being normally distributed.

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

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

::<math>\widehat{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\widehat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left[ (T_{q}^{\widehat{\beta }}\ln ({{T}_{q}})-S_{q}^{\widehat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln ({{X}_{i}}{{}_{q}})}</math>

All variance can be calculated using the Fisher Information Matrix.
<br>
<math>\Lambda </math> is the natural log-likelihood function.

::<math>\Lambda =\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ {{N}_{q}}(\ln (\lambda )+\ln (\beta ))-\lambda (T_{q}^{\beta }-S_{q}^{\beta })+(\beta -1)\underset{i=1}{\overset{{{N}_{q}}}{\mathop \sum }}\,\ln ({{x}_{iq}}) \right]</math>

::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\lambda }^{2}}}</math>

::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }\ln ({{T}_{q}})-S_{q}^{\beta }\ln ({{S}_{q}}) \right]</math>

::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\beta }^{2}}}-\lambda \underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }{{(\ln ({{T}_{q}}))}^{2}}-S_{q}^{\beta }{{(\ln ({{S}_{q}}))}^{2}} \right]</math>

=====Crow Bounds=====
Calculate the conditional maximum likelihood estimate of <math>\tilde{\beta }</math> :

::<math>\tilde{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{iq}}} \right)}</math>

The Crow 2-sided <math>(1-a)</math> 100-percent confidence bounds on <math>\beta </math> are:

::<math>\begin{align}
& {{\beta }_{L}}= & \tilde{\beta }\frac{\chi _{\tfrac{\alpha }{2},2M}^{2}}{2M} \\
& {{\beta }_{U}}= & \tilde{\beta }\frac{\chi _{1-\tfrac{\alpha }{2},2M}^{2}}{2M}
\end{align}</math>

Template:Bounds on beta rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on .CE.B2

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

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