Template:Bounds on growth potential failure intensity rga: Difference between revisions

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(Created page with '===Bounds on Growth Potential Failure Intensity=== ====Fisher Matrix Bounds==== If there are no BC failure modes, the growth potential failure intensity is <math>{{\widehat{r}}_…')
 
(Redirected page to Crow Extended Confidence Bounds)
 
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===Bounds on Growth Potential Failure Intensity===
#REDIRECT [[Crow Extended Confidence Bounds]]
====Fisher Matrix Bounds====
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> .
 
:Then:
 
::<math>\begin{align}
  & 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] \\
& \le  & \frac{1}{T}\left[ \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right] \\
& = & \frac{{{r}_{GP}}}{T} 
\end{align}</math>
 
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:
 
::<math>\sqrt{T}\left( \frac{{{{\hat{r}}}_{GP}}-{{r}_{GP}}}{\sqrt{{{r}_{GP}}}} \right)\sim N(0,1)</math>
 
 
The confidence bounds on the growth potential failure intensity are as follows:
 
::<math>\begin{align}
  & {{r}_{L}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& {{r}_{U}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{{{C}^{4}}}{4}} 
\end{align}</math>
 
where  <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .
<br>
 
====Crow Bounds====
The Crow bounds for the growth potential failure intensity are the same as the Fisher Matrix bounds.  
<br>

Latest revision as of 05:42, 25 August 2012

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