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| ===Bounds on Cumulative MTBF===
| | #REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Cumulative_MTBF_2]] |
| ====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 treated as being normally distributed as well.
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| ::<math>\frac{\ln {{{\hat{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \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={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}}</math>
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| :where:
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| <br>
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| ::<math>{{\hat{m}}_{c}}(t)=\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}</math>
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| ::<math>\begin{align}
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| & Var({{{\hat{m}}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\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(\hat{\beta },\hat{\lambda })\,
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| \end{align}</math>
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| The variance calculation is the same as Eqn. (variances) and:
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| ::<math>\begin{align}
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| & \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}\ln t \\
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| & \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}}{{t}^{1-\hat{\beta }}}
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| \end{align}</math>
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| ====Crow Bounds====
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| 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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