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

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(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…')
 
 
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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:
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::<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>

Latest revision as of 04:03, 24 August 2012

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