|
|
| 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>
| |