|
|
| Line 1: |
Line 1: |
| ===Bounds on Time Given Cumulative MTBF===
| | #REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Time_Given_Cumulative_MTBF]] |
| ====Fisher Matrix Bounds====
| |
| The time, <math>T</math> , must be positive, thus <math>\ln T</math> is treated as being normally distributed.
| |
| ::<math>\frac{\ln \hat{T}-\ln T}{\sqrt{Var(\ln \hat{T}})}\ \tilde{\ }\ N(0,1)</math>
| |
| <br>
| |
| Confidence bounds on the time are given by:
| |
| <br>
| |
| ::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}</math>
| |
| :where:
| |
| ::<math>\begin{align}
| |
| & Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
| |
| & & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda })
| |
| \end{align}</math>
| |
| <br>
| |
| The variance calculation is the same as Eqn. (variance1) and:
| |
| <br>
| |
| ::<math>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}</math>
| |
| | |
| ::<math>\begin{align}
| |
| & \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot \,{{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot \text{ }{{m}_{c}})}{{{(1-\beta )}^{2}}} \\
| |
| & \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \text{ }\cdot \text{ }{{m}_{c}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}
| |
| \end{align}</math>
| |
| | |
| ====Crow Bounds====
| |
| :Step 1: Calculate <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}</math> .
| |
| :Step 2: Use the equations from 5.2.8.2 to calculate the bounds on time given the cumulative failure intensity.
| |