Template:Bounds on lambda camsaa-gd - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.BB 2

2012-08-24T03:45:46Z

Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.BB 2

← Older revision		Revision as of 03:45, 24 August 2012
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	~~===Bounds on <math>\lambda </math>===~~		#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on__.CE.BB_2]]
	~~====Fisher Matrix Bounds====~~
	~~The parameter <math>\lambda </math> must be positive, thus <math>\ln \lambda </math> is treated as being normally distributed as well. These bounds are based on:~~

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

	~~The approximate confidence bounds on <math>\lambda </math> are given as:~~

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

	~~:where:~~

	~~::<math>\hat{\lambda }=\frac{n}{T_{k}^{{\hat{\beta }}}}</math>~~

	~~The variance calculation is the same as Eqn~~. ~~(variances)~~.

	~~====Crow Bounds====~~
	~~<br>~~
	~~'''Time Terminated Data'''~~
	~~<br>~~
	~~For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval, the confidence bounds on <math>\lambda </math> are:~~

	~~::<math>\begin{align}~~
	~~& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\~~
	~~& {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot T_{k}^{\beta }}~~
	~~\end{align}</math>~~

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

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

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

2012-01-05T23:07:24Z

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

New page

===Bounds on <math>\lambda </math>===
====Fisher Matrix Bounds====
The parameter <math>\lambda </math> must be positive, thus <math>\ln \lambda </math> is treated as being normally distributed as well. These bounds are based on:

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

The approximate confidence bounds on <math>\lambda </math> are given as:

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

:where:

::<math>\hat{\lambda }=\frac{n}{T_{k}^{{\hat{\beta }}}}</math>

The variance calculation is the same as Eqn. (variances).

====Crow Bounds====
<br>
'''Time Terminated Data'''
<br>
For the 2-sided <math>(1-\alpha )</math> 100-percent confidence interval, the confidence bounds on <math>\lambda </math> are:

::<math>\begin{align}
& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\
& {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot T_{k}^{\beta }}
\end{align}</math>

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

::<math>\begin{align}
& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\
& {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }}
\end{align}</math>

Template:Bounds on lambda camsaa-gd - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on .CE.BB 2

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

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