Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

2012-08-25T05:42:52Z

Redirected page to Crow Extended Confidence Bounds

← Older revision		Revision as of 05:42, 25 August 2012
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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>~~

Nicolette Young: 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 ${{\widehat{r}}_…'$

2012-01-10T20:52:41Z

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}}_…'

New page

===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}}_{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>

Template:Bounds on growth potential failure intensity rga - Revision history

Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

Nicolette Young: 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 {{\widehat{r}}_…'

Nicolette Young: 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 ${{\widehat{r}}_…'$