Template:Bounds on cumulative number of failures camsaa-gd: Difference between revisions

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===Bounds on Cumulative Number of Failures===
#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Cumulative_Number_of_Failures_2]]
====Fisher Matrix Bounds====
The cumulative number of failures,  <math>N(t)</math> , must be positive, thus  <math>\ln N(t)</math>  is treated as being normally distributed. 
 
::<math>\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)</math>
 
::<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}</math>
 
:where:
 
::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
 
 
::<math>\begin{align}
  & Var(\hat{N}(t))= & {{\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
&  & +2\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) 
\end{align}</math>
 
 
The variance calculation is the same as Eqn. (variances) and: 
 
::<math>\begin{align}
  & \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\
& \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}} 
\end{align}</math>
 
 
====Crow Bounds====
The Crow confidence bounds on cumulative number of failures are:
 
::<math>\begin{align}
  & {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\
& {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}} 
\end{align}</math>
 
 
where  <math>{{\lambda }_{i}}{{(T)}_{L}}</math>  and  <math>{{\lambda }_{i}}{{(T)}_{U}}</math>  can be obtained from Eqn. (dsaf).
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Latest revision as of 03:53, 24 August 2012

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