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| ===Bounds on Demonstrated Failure Intensity===
| | #REDIRECT [[Crow Extended Confidence Bounds]] |
| ====Fisher Matrix Bounds====
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| If there are no BC failure modes, the demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{BD}}}{T}</math> . Thus:
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| ::<math>Var({{\hat{\lambda }}_{D}}(t))=\frac{{{N}_{A}}}{{{T}^{2}}}+\frac{{{N}_{BD}}}{{{T}^{2}}}=\frac{{{\lambda }_{D}}(t)}{T}</math>
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| <br>
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| :and:
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| ::<math>\sqrt{T}\left( \frac{{{{\hat{\lambda }}}_{D}}(T)-{{\lambda }_{D}}(T)}{\sqrt{{{\lambda }_{D}}(T)}} \right)\sim N(0,1)</math>
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| ::<math>{{\lambda }_{D}}(T)={{\hat{\lambda }}_{D}}(T)+\frac{{{C}^{2}}}{2}\pm \sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}}</math>
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| where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .
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| <br>
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| If there are BC failure modes, the demonstrated failure intensity, <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}</math> , is actually the instantaneous failure intensity based on all of the data. <math>{{\lambda }_{CA}}(T)</math> must be positive, thus <math>\ln {{\lambda }_{CA}}(T)</math> is approximately treated as being normally distributed.
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| ::<math>\frac{\ln {{{\hat{\lambda }}}_{CA}}(T)-\ln {{\lambda }_{CA}}(T)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{CA}}(T)})}\sim 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 }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}}</math>
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| where <math>{{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}</math> .
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| ::<math>\begin{align}
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| & Var({{{\hat{\lambda }}}_{CA}}(T))= & {{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
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| & & +2\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{CA}}(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 described in Chapter 5.
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| <br>
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| ====Crow Bounds====
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| <br>
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| If there are no BC failure modes then:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{D}}(T)]}_{l}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\
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| & {{[{{\lambda }_{D}}(T)]}_{u}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N}
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| \end{align}</math>
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| where <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}</math> .
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| <br>
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| If there are BC modes then the confidence bounds on the demonstrated failure intensity are calculated as presented in Chapter 5.
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| <br>
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