Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Reliability

2012-08-27T00:38:10Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Reliability

← Older revision		Revision as of 00:38, 27 August 2012
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	~~====Bounds on Reliability====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Reliability]]
	~~=====Fisher Matrix Bounds=====~~
	~~These bounds are based on:~~

	~~::<math>\log it(\widehat{R}(t))\sim N(0,1)</math>~~


	~~::<math>\log it(\widehat{R}(t))=\ln \left\{ \frac{\widehat{R}(t)}{1-\widehat{R}(t)} \right\}</math>~~


	~~The confidence bounds on reliability are given by:~~

	~~::<math>CB=\frac{\widehat{R}(t)}{\widehat{R}(t)+(1-\widehat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{R}(t))}/\left~~[ ~~\widehat{R}(t)(1-\widehat{R}(t)) \right]}}}</math>~~


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


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

	~~::<math>\begin{align}~~
	& \frac{\partial R}{\partial \beta }= & {{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}[\lambda {{t}^{\widehat{\beta }}}\ln (t)-\lambda {{(t+d)}^{\widehat{\beta }}}\ln (t+d)] \\
	~~& \frac{\partial R}{\partial \lambda }= & {{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}[{{t}^{\widehat{\beta }}}-{{(t+d)}^{\widehat{\beta }}}]~~
	~~\end{align}</math>~~


	~~=====Crow Bounds=====~~
	~~''Failure Terminated Data''~~
	~~<br>~~
	~~With failure terminated data, the 100( <math>1-\alpha </math> )% confidence interval for the current reliability at time <math>t</math> in a specified mission time <math>d</math> is:~~

	~~::<math>({{[\widehat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{~~[~~\hat{R}(d)~~]~~}^{\tfrac{1}{{{p}_{2}}}}})</math>~~

	~~:where~~

	~~::<math>\widehat{R}(\tau )={{e}^{-[\widehat{\lambda }{{(t+\tau )}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}</math>~~

	~~<math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> can be obtained from Eqn. (ft).~~
	~~<br>~~
	~~<br>~~
	~~''Time Terminated Data''~~
	~~<br>~~
	~~With time terminated data, the 100( <math>1-\alpha </math> )% confidence interval for the current reliability at time <math>t</math> in a specified mission time <math>\tau </math> is:~~

	~~::<math>({{[\widehat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})</math>~~

	~~:where:~~

	~~::<math>\widehat{R}(d)={{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}~~]~~}}</math>~~

	~~<math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> can be obtained from Eqn. (tt).~~

Nicolette Young: Created page with '====Bounds on Reliability==== =====Fisher Matrix Bounds===== These bounds are based on: :: $\log it(\widehat{R}(t))\sim N(0,1)$ :: $\log it(\widehat{R}(t))=\ln …'$

2012-01-10T22:47:06Z

Created page with '====Bounds on Reliability==== =====Fisher Matrix Bounds===== These bounds are based on: ::<math>\log it(\widehat{R}(t))\sim N(0,1)</math> ::<math>\log it(\widehat{R}(t))=\ln …'

New page

====Bounds on Reliability====
=====Fisher Matrix Bounds=====
These bounds are based on:

::<math>\log it(\widehat{R}(t))\sim N(0,1)</math>

::<math>\log it(\widehat{R}(t))=\ln \left\{ \frac{\widehat{R}(t)}{1-\widehat{R}(t)} \right\}</math>

The confidence bounds on reliability are given by:

::<math>CB=\frac{\widehat{R}(t)}{\widehat{R}(t)+(1-\widehat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{R}(t))}/\left[ \widehat{R}(t)(1-\widehat{R}(t)) \right]}}}</math>

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

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

::<math>\begin{align}
& \frac{\partial R}{\partial \beta }= & {{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}[\lambda {{t}^{\widehat{\beta }}}\ln (t)-\lambda {{(t+d)}^{\widehat{\beta }}}\ln (t+d)] \\
& \frac{\partial R}{\partial \lambda }= & {{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}[{{t}^{\widehat{\beta }}}-{{(t+d)}^{\widehat{\beta }}}]
\end{align}</math>

=====Crow Bounds=====
''Failure Terminated Data''
<br>
With failure terminated data, the 100( <math>1-\alpha </math> )% confidence interval for the current reliability at time <math>t</math> in a specified mission time <math>d</math> is:

::<math>({{[\widehat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})</math>

:where

::<math>\widehat{R}(\tau )={{e}^{-[\widehat{\lambda }{{(t+\tau )}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}</math>

<math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> can be obtained from Eqn. (ft).
<br>
<br>
''Time Terminated Data''
<br>
With time terminated data, the 100( <math>1-\alpha </math> )% confidence interval for the current reliability at time <math>t</math> in a specified mission time <math>\tau </math> is:

::<math>({{[\widehat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})</math>

:where:

::<math>\widehat{R}(d)={{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}</math>

<math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> can be obtained from Eqn. (tt).

Template:Bounds on reliability rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Reliability

Nicolette Young: Created page with '====Bounds on Reliability==== =====Fisher Matrix Bounds===== These bounds are based on: ::\log it(\widehat{R}(t))\sim N(0,1) ::\log it(\widehat{R}(t))=\ln …'

Nicolette Young: Created page with '====Bounds on Reliability==== =====Fisher Matrix Bounds===== These bounds are based on: :: $\log it(\widehat{R}(t))\sim N(0,1)$ :: $\log it(\widehat{R}(t))=\ln …'$