Template:Bounds on instantaneous mtbf rsa: Difference between revisions

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(Created page with '====Bounds on Instantaneous MTBF==== =====Fisher Matrix Bounds===== The instantaneous MTBF, <math>{{m}_{i}}(t)</math> , must be positive, thus <math>\ln {{m}_{i}}(t)</math> is…')
 
 
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====Bounds on Instantaneous MTBF====
#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Instantaneous_MTBF]]
=====Fisher Matrix Bounds=====
The instantaneous MTBF,  <math>{{m}_{i}}(t)</math> , must be positive, thus  <math>\ln {{m}_{i}}(t)</math>  is approximately treated as being normally distributed.
 
::<math>\frac{\ln ({{\widehat{m}}_{i}}(t))-\ln ({{m}_{i}}(t))}{\sqrt{Var\left[ \ln ({{\widehat{m}}_{i}}(t)) \right]}}\ \tilde{\ }\ N(0,1)</math>
 
 
The approximate confidence bounds on the instantaneous MTBF are then estimated from:
 
::<math>CB={{\widehat{m}}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\widehat{m}}_{i}}(t))}/{{\widehat{m}}_{i}}(t)}}</math>
 
:where:
 
::<math>{{\widehat{m}}_{i}}(t)=\frac{1}{\lambda \beta {{t}^{\beta -1}}}</math>
 
::<math>\begin{align}
  & Var({{\widehat{m}}_{i}}(t))= & {{\left( \frac{\partial {{m}_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial {{m}_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
&  & +2\left( \frac{\partial {{m}_{i}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{i}}(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda }) 
\end{align}</math>
 
 
The variance calculation is the same as (var1), (var2) and (var3).
 
::<math>\begin{align}
  & \frac{\partial {{m}_{i}}(t)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }{{\widehat{\beta }}^{2}}}{{t}^{1-\widehat{\beta }}}-\frac{1}{\widehat{\lambda }\widehat{\beta }}{{t}^{1-\widehat{\beta }}}\ln (t) \\
& \frac{\partial {{m}_{i}}(t)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}\widehat{\beta }}{{t}^{1-\widehat{\beta }}} 
\end{align}</math>
 
 
=====Crow Bounds=====
''Failure Terminated Data''
<br>
To calculate the bounds for failure terminated data, consider the following equation:
 
::<math>G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx</math>
 
Find the values  <math>{{p}_{1}}</math>  and  <math>{{p}_{2}}</math>  by finding the solution  <math>c</math>  to  <math>G({{n}^{2}}/c|n)=\xi </math>  for  <math>\xi =\tfrac{\alpha }{2}</math>  and  <math>\xi =1-\tfrac{\alpha }{2}</math> , respectively. If using the biased parameters,  <math>\hat{\beta }</math>  and  <math>\hat{\lambda }</math> , then the upper and lower confidence bounds are:
 
::<math>\begin{align}
  & {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{p}_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{p}_{2}} 
\end{align}</math>
 
where  <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> . If using the unbiased parameters,  <math>\bar{\beta }</math>  and  <math>\bar{\lambda }</math> , then the upper and lower confidence bounds are:
 
::<math>\begin{align}
  & {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}} 
\end{align}</math>
 
where  <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> .
<br>
<br>
''Time Terminated Data''
<br>
To calculate the bounds for time terminated data, consider the following equation where  <math>{{I}_{1}}(.)</math>  is the modified Bessel function of order one:
 
::<math>H(x|k)=\underset{j=1}{\overset{k}{\mathop \sum }}\,\frac{{{x}^{2j-1}}}{{{2}^{2j-1}}(j-1)!j!{{I}_{1}}(x)}</math>
 
Find the values  <math>{{\Pi }_{1}}</math>  and  <math>{{\Pi }_{2}}</math>  by finding the solution  <math>x</math>  to  <math>H(x|k)=\tfrac{\alpha }{2}</math>  and  <math>H(x|k)=1-\tfrac{\alpha }{2}</math>  in the cases corresponding to the lower and upper bounds, respectively. <br>
Calculate  <math>\Pi =\tfrac{{{n}^{2}}}{4{{x}^{2}}}</math>  for each case. If using the biased parameters,  <math>\hat{\beta }</math>  and  <math>\hat{\lambda }</math> , then the upper and lower confidence bounds are:
 
::<math>\begin{align}
  & {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}} 
\end{align}</math>
 
where  <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> . If using the unbiased parameters,  <math>\bar{\beta }</math>  and  <math>\bar{\lambda }</math> , then the upper and lower confidence bounds are:
 
::<math>\begin{align}
  & {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{2}} 
\end{align}</math>
 
where  <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> .
<br>
<br>

Latest revision as of 00:33, 27 August 2012

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