Template:Bounds on reliability rsa: Difference between revisions

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(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 …')
 
 
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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).

Latest revision as of 00:38, 27 August 2012

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