Template:Bounds on lambda camsaa-cb: Difference between revisions

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(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…')
 
 
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===Bounds on  <math>\lambda </math>===
#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on__.CE.BB]]
====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:
<br>
::<math>\frac{\ln \hat{\lambda }-\ln \lambda }{\sqrt{Var(\ln \hat{\lambda }})}\ \tilde{\ }\ N(0,1)</math>
<br>
The approximate confidence bounds on  <math>\lambda </math>  are given as:
<br>
::<math>C{{B}_{\lambda }}=\hat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}}</math>
<br>
:where:
<br>
::<math>\hat{\lambda }=\frac{n}{{{T}^{*\hat{\beta }}}}</math>
<br>
The variance calculation is the same as Eqn. (variance1).
 
====Crow Bounds====
'''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{{T}^{{\hat{\beta }}}}} \\
& {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2{{T}^{{\hat{\beta }}}}} 
\end{align}</math>
 
The fractiles can be found in the tables of the  <math>{{\chi }^{2}}</math>  distribution.
<br>
<br>
'''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{{T}^{{\hat{\beta }}}}} \\
& {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} 
\end{align}</math>

Latest revision as of 04:05, 24 August 2012

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