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| ====Bounds on <math>\lambda </math>====
| | #REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on__.CE.BB]] |
| =====Fisher Matrix Bounds=====
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| The parameter <math>\lambda </math> must be positive, thus <math>\ln \lambda </math> is approximately treated as being normally distributed. These bounds are based on:
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| ::<math>\frac{\ln (\widehat{\lambda })-\ln (\lambda )}{\sqrt{Var\left[ \ln (\widehat{\lambda }) \right]}}\ \tilde{\ }\ N(0,1)</math>
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| <br>
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| The approximate confidence bounds on <math>\lambda </math> are given as:
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| ::<math>C{{B}_{\lambda }}=\widehat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{\lambda })}/\widehat{\lambda }}}</math>
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| where <math>\widehat{\lambda }=\tfrac{n}{T_{K}^{{\hat{\beta }}}}</math> .
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| The variance calculation is the same as Eqns. (var1), (var2) and (var3).
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| <br>
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| <br>
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| =====Crow Bounds=====
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| ''Time Terminated''
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| <br>
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| The confidence bounds on <math>\lambda </math> for time terminated data are calculated using:
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| ::<math>\begin{align}
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| & {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\
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| & {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}
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| \end{align}</math>
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| ''Failure Terminated''
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| <br>
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| The confidence bounds on <math>\lambda </math> for failure terminated data are calculated using:
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| ::<math>\begin{align}
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| & {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\
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| & {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}
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| \end{align}</math>
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