Template:Bounds on beta rsa: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
(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 approximately treate…')
 
 
Line 1: Line 1:
====Bounds on  <math>\beta </math>====
#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on__.CE.B2]]
=====Fisher Matrix Bounds=====
The parameter  <math>\beta </math>  must be positive, thus  <math>\ln \beta </math>  is approximately treated as being normally distributed.
 
 
::<math>\frac{\ln (\widehat{\beta })-\ln (\beta )}{\sqrt{Var\left[ \ln (\widehat{\beta }) \right]}}\ \tilde{\ }\ N(0,1)</math>
 
 
::<math>C{{B}_{\beta }}=\widehat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{\beta })}/\widehat{\beta }}}</math>
 
 
::<math>\widehat{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\widehat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left[ (T_{q}^{\widehat{\beta }}\ln ({{T}_{q}})-S_{q}^{\widehat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln ({{X}_{i}}{{}_{q}})}</math>
 
 
All variance can be calculated using the Fisher Information Matrix.
<br>
<math>\Lambda </math>  is the natural log-likelihood function.
 
 
::<math>\Lambda =\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ {{N}_{q}}(\ln (\lambda )+\ln (\beta ))-\lambda (T_{q}^{\beta }-S_{q}^{\beta })+(\beta -1)\underset{i=1}{\overset{{{N}_{q}}}{\mathop \sum }}\,\ln ({{x}_{iq}}) \right]</math>
 
 
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\lambda }^{2}}}</math>
 
 
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }\ln ({{T}_{q}})-S_{q}^{\beta }\ln ({{S}_{q}}) \right]</math>
 
 
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\beta }^{2}}}-\lambda \underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }{{(\ln ({{T}_{q}}))}^{2}}-S_{q}^{\beta }{{(\ln ({{S}_{q}}))}^{2}} \right]</math>
 
=====Crow Bounds=====
Calculate the conditional maximum likelihood estimate of  <math>\tilde{\beta }</math> :
 
 
::<math>\tilde{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{iq}}} \right)}</math>
 
 
The Crow 2-sided  <math>(1-a)</math> 100-percent confidence bounds on  <math>\beta </math>  are:
 
::<math>\begin{align}
  & {{\beta }_{L}}= & \tilde{\beta }\frac{\chi _{\tfrac{\alpha }{2},2M}^{2}}{2M} \\
& {{\beta }_{U}}= & \tilde{\beta }\frac{\chi _{1-\tfrac{\alpha }{2},2M}^{2}}{2M} 
\end{align}</math>

Latest revision as of 00:30, 27 August 2012

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Bounds_on_beta_rsa&oldid=33915"