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| ====Bounds on Reliability====
| | #REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Reliability]] |
| =====Fisher Matrix Bounds=====
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| These bounds are based on:
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| ::<math>\log it(\widehat{R}(t))\sim N(0,1)</math>
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| ::<math>\log it(\widehat{R}(t))=\ln \left\{ \frac{\widehat{R}(t)}{1-\widehat{R}(t)} \right\}</math>
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| The confidence bounds on reliability are given by:
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| ::<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>
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| ::<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>
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| The variance calculation is the same as Eqns. (var1), (var2) and (var3).
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| ::<math>\begin{align}
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| & \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)] \\
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| & \frac{\partial R}{\partial \lambda }= & {{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}[{{t}^{\widehat{\beta }}}-{{(t+d)}^{\widehat{\beta }}}]
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| \end{align}</math>
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| =====Crow Bounds=====
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| ''Failure Terminated Data''
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| <br>
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| 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:
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| ::<math>({{[\widehat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})</math>
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| :where
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| ::<math>\widehat{R}(\tau )={{e}^{-[\widehat{\lambda }{{(t+\tau )}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}</math>
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| <math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> can be obtained from Eqn. (ft).
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| <br>
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| <br>
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| ''Time Terminated Data''
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| <br>
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| 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:
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| ::<math>({{[\widehat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})</math>
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| :where:
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| ::<math>\widehat{R}(d)={{e}^{-[\widehat{\lambda }{{(t+d)}^{\widehat{\beta }}}-\widehat{\lambda }{{t}^{\widehat{\beta }}}]}}</math>
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| <math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> can be obtained from Eqn. (tt).
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