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| ====Bounds on Time Given Cumulative Failure Intensity====
| | #REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Time_Given_Cumulative_Failure_Intensity]] |
| =====Fisher Matrix Bounds=====
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| The time, <math>T</math> , must be positive, thus <math>\ln T</math> is approximately treated as being normally distributed.
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| ::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ \ln \widehat{T} \right]}}\ \tilde{\ }\ N(0,1)</math>
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| The confidence bounds on the time are given by:
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| ::<math>CB=\widehat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{T})}/\widehat{T}}}</math>
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| :where:
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| ::<math>Var(\widehat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\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>\widehat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}</math>
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| ::<math>\begin{align}
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| & \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\
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| & \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )}
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| \end{align}</math>
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| =====Crow Bounds=====
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| Step 1: Calculate:
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| ::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}</math>
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| Step 2: Estimate the number of failures:
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| ::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}</math>
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| Step 3: Obtain the confidence bounds on time given the cumulative failure intensity by solving for <math>{{t}_{l}}</math> and <math>{{t}_{u}}</math> in the following equations:
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| ::<math>\begin{align}
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| & {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\
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| & {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)}
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| \end{align}</math>
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