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

Latest revision as of 00:31, 27 August 2012

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