Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.B2

2012-08-24T04:03:30Z

Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.B2

← Older revision		Revision as of 04:03, 24 August 2012
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			#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on__.CE.B2]]
	~~===Bounds on <math>\beta </math>===~~
	~~====Fisher Matrix Bounds====~~
	~~The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normally distributed as well.~~

	~~<br>~~
	~~::<math>\frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1)</math>~~

	~~<br>~~
	~~The approximate confidence bounds are given as:~~

	~~<br>~~
	~~::<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}</math>~~

	~~<br>~~
	<math>\alpha </math> in <math>{{z}_{\alpha }}</math> is different ( <math>\alpha /2</math> , <math>\alpha </math> ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.

	~~<br>~~
	~~::<math>\left~~[ ~~\begin{matrix}~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\~~
	~~\end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left~~[ ~~\begin{matrix}~~
	~~Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\~~
	~~Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\~~
	~~\end{matrix} \right]</math>~~
	~~<br>~~
	~~::<math>\Lambda </math> is the natural log-likelihood function:~~
	~~<br>~~
	~~::<math>\Lambda =N\ln \lambda +N\ln \beta -\lambda {{T}^{\beta }}+(\beta -1)\underset{i=1}{\overset{N}{\mathop \sum }}\,\ln {{T}_{i}}</math>~~
	~~<br>~~
	~~::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}}</math>~~
	~~<br>~~
	~~:and:~~
	~~<br>~~
	~~::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}}</math>~~
	~~<br>~~
	~~:also:~~
	~~<br>~~
	~~::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T</math>~~

	~~====~~Crow ~~Bounds====~~
	~~'''Time Terminated Data'''~~

	~~For the 2~~-~~sided <math>(1~~-~~\alpha )</math> 100-percent confidence interval on <math>\beta </math> , calculate:~~

	~~::<math>\begin{align}~~
	~~& {{D}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2(N-1)} \\~~
	~~& {{D}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2(N-1)}~~
	~~\end{align}</math>~~

	~~The fractiles can be found in the tables of the <math>{{\chi }^{2}}</math> distribution~~. ~~Thus the confidence bounds on <math>\beta </math> are:~~

	~~::<math>\begin{align}~~
	~~& {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\~~
	~~& {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }~~
	~~\end{align}</math>~~

	~~<br>~~
	~~'''Failure Terminated Data'''~~
	~~<br>~~
	~~For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>\beta </math> , calculate:~~

	~~::<math>\begin{align}~~
	~~& {{D}_{L}}= & \frac{N\cdot \chi _{\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)} \\~~
	~~& {{D}_{U}}= & \frac{N\cdot \chi _{1-\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)}~~
	~~\end{align}</math>~~

	~~Thus the confidence bounds on <math>\beta </math> are:~~

	~~::<math>\begin{align}~~
	~~& {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\~~
	~~& {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }~~
	~~\end{align}</math>~~

Nicolette Young: Created page with ' ===Bounds on $\beta$ === ====Fisher Matrix Bounds==== The parameter $\beta$ must be positive, thus $\ln \beta$ is treated as being normal…'

2012-01-05T21:06:48Z

Created page with ' ===Bounds on <math>\beta </math>=== ====Fisher Matrix Bounds==== The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normal…'

New page

===Bounds on <math>\beta </math>===
====Fisher Matrix Bounds====
The parameter <math>\beta </math> must be positive, thus <math>\ln \beta </math> is treated as being normally distributed as well.

 
::<math>\frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1)</math>

 
The approximate confidence bounds are given as:

 
::<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}</math>

 
<math>\alpha </math> in <math>{{z}_{\alpha }}</math> is different ( <math>\alpha /2</math> , <math>\alpha </math> ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.

 
::<math>\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\
\end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left[ \begin{matrix}
Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\
Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\
\end{matrix} \right]</math>
 
::<math>\Lambda </math> is the natural log-likelihood function:
 
::<math>\Lambda =N\ln \lambda +N\ln \beta -\lambda {{T}^{\beta }}+(\beta -1)\underset{i=1}{\overset{N}{\mathop \sum }}\,\ln {{T}_{i}}</math>
 
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}}</math>
 
:and:
 
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}}</math>
 
:also:
 
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T</math>

====Crow Bounds====
'''Time Terminated Data'''

For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>\beta </math> , calculate:

::<math>\begin{align}
& {{D}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2(N-1)} \\
& {{D}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2(N-1)}
\end{align}</math>

The fractiles can be found in the tables of the <math>{{\chi }^{2}}</math> distribution. Thus the confidence bounds on <math>\beta </math> are:

::<math>\begin{align}
& {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\
& {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }
\end{align}</math>

 
'''Failure Terminated Data'''
 
For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>\beta </math> , calculate:

::<math>\begin{align}
& {{D}_{L}}= & \frac{N\cdot \chi _{\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)} \\
& {{D}_{U}}= & \frac{N\cdot \chi _{1-\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)}
\end{align}</math>

Thus the confidence bounds on <math>\beta </math> are:

::<math>\begin{align}
& {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\
& {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }
\end{align}</math>

Template:Bounds on beta camsaa-cb - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.B2

Nicolette Young: Created page with ' ===Bounds on \beta === ====Fisher Matrix Bounds==== The parameter \beta must be positive, thus \ln \beta is treated as being normal…'

Nicolette Young: Created page with ' ===Bounds on $\beta$ === ====Fisher Matrix Bounds==== The parameter $\beta$ must be positive, thus $\ln \beta$ is treated as being normal…'