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| ====Bounds on Instantaneous Failure Intensity====
| | #REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Instantaneous_Failure_Intensity]] |
| =====Fisher Matrix Bounds=====
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| The instantaneous failure intensity, <math>{{\lambda }_{i}}(t)</math> , must be positive, thus <math>\ln {{\lambda }_{i}}(t)</math> is approximately treated as being normally distributed.
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| ::<math>\frac{\ln ({{\widehat{\lambda }}_{i}}(t))-\ln ({{\lambda }_{i}}(t))}{\sqrt{Var\left[ \ln ({{\widehat{\lambda }}_{i}}(t)) \right]}}\sim N(0,1)</math>
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| <br>
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| The approximate confidence bounds on the instantaneous failure intensity are then estimated from:
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| ::<math>CB={{\widehat{\lambda }}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\widehat{\lambda }}_{i}}(t))}/{{\widehat{\lambda }}_{i}}(t)}}</math>
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| where <math>{{\lambda }_{i}}(t)=\lambda \beta {{t}^{\beta -1}}</math> and:
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| ::<math>\begin{align}
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| & Var({{\widehat{\lambda }}_{i}}(t))= & {{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
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| & & +2\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })
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| \end{align}</math>
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| <br>
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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 {{\lambda }_{i}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\widehat{\beta }-1}}+\hat{\lambda }\hat{\beta }{{t}^{\widehat{\beta }-1}}\ln (t) \\
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| & \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda }= & \widehat{\beta }{{t}^{\widehat{\beta }-1}}
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| \end{align}</math>
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| =====Crow Bounds=====
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| The Crow instantaneous failure intensity confidence bounds are given as:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\
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| & {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}}
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| \end{align}</math>
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