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| ====Bounds on Cumulative MTBF====
| | #REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Cumulative_MTBF]] |
| =====Fisher Matrix Bounds=====
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| The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is approximately treated as being normally distributed.
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| ::<math>\frac{\ln ({{\widehat{m}}_{c}}(t))-\ln ({{m}_{c}}(t))}{\sqrt{Var\left[ \ln ({{\widehat{m}}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)</math>
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| The approximate confidence bounds on the cumulative MTBF are then estimated from:
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| ::<math>CB={{\widehat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\widehat{m}}_{c}}(t))}/{{\widehat{m}}_{c}}(t)}}</math>
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| :where:
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| ::<math>{{\widehat{m}}_{c}}(t)=\frac{1}{\widehat{\lambda }}{{t}^{1-\widehat{\beta }}}</math>
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| ::<math>\begin{align}
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| & Var({{\widehat{m}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
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| & & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })\,
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| \end{align}</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 {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }}{{t}^{1-\widehat{\beta }}}\ln (t) \\
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| & \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}}{{t}^{1-\widehat{\beta }}}
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| \end{align}</math>
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| =====Crow Bounds=====
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| To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:
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| ::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}</math>
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| ::<math>C{{(t)}_{u}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}</math>
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| :Then
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| ::<math>\begin{align}
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| & {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\
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| & {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}
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| \end{align}</math>
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