Template:Bounds on lambda camsaa-gd: 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_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>

Latest revision as of 03:45, 24 August 2012

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