|
|
| Line 1: |
Line 1: |
| ===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).
| |
| <br>
| |