Template:Bounds on time given instantaneous failure intensity rsa: Difference between revisions

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(Created page with '====Bounds on Time Given Instantaneous Failure Intensity==== =====Fisher Matrix Bounds===== These bounds are based on: ::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ …')
 
 
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====Bounds on Time Given Instantaneous Failure Intensity====
#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Time_Given_Instantaneous_Failure_Intensity]]
=====Fisher Matrix Bounds=====
These bounds are based on:
 
::<math>\frac{\ln (\widehat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\widehat{T}) \right]}}\sim N(0,1)</math>
 
 
The confidence bounds on the time are given by:
 
 
::<math>CB=\widehat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{T})}/\widehat{T}}}</math>
 
:where:
 
::<math>\begin{align}
  & Var(\widehat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
&  & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda }) 
\end{align}</math>
 
The variance calculation is the same as Eqns. (var1), (var2) and (var3).
 
::<math>\widehat{T}={{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}</math>
 
 
::<math>\begin{align}
  & \frac{\partial T}{\partial \beta }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}[-\frac{\ln (\tfrac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta })}{{{(\beta -1)}^{2}}}+\frac{1}{\beta (1-\beta )}] \\
& \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )} 
\end{align}</math>
 
 
=====Crow Bounds=====
Step 1: Calculate  <math>{{\lambda }_{i}}(T)=\tfrac{1}{MTB{{F}_{i}}}</math> .
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Step 2: Use the equations from 13.1.7.9 to calculate the bounds on time given the instantaneous failure intensity.
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Latest revision as of 00:37, 27 August 2012

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