Template:Bounds on cumulative number of failures camsaa-gd

Fisher Matrix Bounds
The cumulative number of failures, $$N(t)$$, must be positive, thus  $$\ln N(t)$$  is treated as being normally distributed.


 * $$\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)$$


 * $$N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}$$


 * where:


 * $$\hat{N}(t)=\hat{\lambda }{{t}^}$$


 * $$\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}$$

The variance calculation is the same as Eqn. (variances) and:


 * $$\begin{align}

& \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^}\ln t \\ & \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^} \end{align}$$

Crow Bounds
The Crow confidence bounds on cumulative number of failures are:


 * $$\begin{align}

& {{N}_{L}}(T)= & \frac{T}{{\lambda }_{i}}{{(T)}_{L}} \\ & {{N}_{U}}(T)= & \frac{T}{{\lambda }_{i}}{{(T)}_{U}} \end{align}$$

where $${{\lambda }_{i}}{{(T)}_{L}}$$  and  $${{\lambda }_{i}}{{(T)}_{U}}$$  can be obtained from Eqn. (dsaf).