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| ===Bounds on Instantaneous Failure Intensity===
| | #REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Instantaneous_Failure_Intensity_2]] |
| ====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 treated as being normally distributed.
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| ::<math>\frac{\ln {{{\hat{\lambda }}}_{i}}(t)-\ln {{\lambda }_{i}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{i}}(t)})}\tilde{\ }N(0,1)</math>
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| The approximate confidence bounds on the instantaneous failure intensity are then estimated from:
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| ::<math>CB={{\hat{\lambda }}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{i}}(t))}/{{{\hat{\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({{{\hat{\lambda }}}_{i}}(t))= & {{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\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(\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 {{\lambda }_{i}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}+\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}\ln t \\
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| & \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda }= & \hat{\beta }{{t}^{\hat{\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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