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| ===Bounds on Growth Potential Failure Intensity===
| | #REDIRECT [[Crow Extended Confidence Bounds]] |
| ====Fisher Matrix Bounds====
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| If there are no BC failure modes, the growth potential failure intensity is <math>{{\widehat{r}}_{GP}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}</math> .
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| :Then:
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| ::<math>\begin{align}
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| & Var({{\widehat{r}}_{GP}})= & \frac{1}{T}\left[ \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,{{(1-{{d}_{i}})}^{2}}\frac{{{N}_{i}}}{T} \right] \\
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| & \le & \frac{1}{T}\left[ \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right] \\
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| & = & \frac{{{r}_{GP}}}{T}
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| \end{align}</math>
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| If there are BC failure modes, the growth potential failure intensity is <math>{{\widehat{r}}_{GP}}(T)={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T},</math> <math>Var({{\widehat{r}}_{GP}})\approx \tfrac{{{r}_{GP}}}{T}</math> . Therefore:
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| ::<math>\sqrt{T}\left( \frac{{{{\hat{r}}}_{GP}}-{{r}_{GP}}}{\sqrt{{{r}_{GP}}}} \right)\sim N(0,1)</math>
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| The confidence bounds on the growth potential failure intensity are as follows:
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| ::<math>\begin{align}
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| & {{r}_{L}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
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| & {{r}_{U}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{{{C}^{4}}}{4}}
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| \end{align}</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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| ====Crow Bounds====
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| The Crow bounds for the growth potential failure intensity are the same as the Fisher Matrix bounds.
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| <br>
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