Template:Bounds on instantaneous failure intensity camsaa-cb

Fisher Matrix Bounds
The instantaneous failure intensity, $${{\lambda }_{i}}(t)$$, must be positive, thus  $$\ln {{\lambda }_{i}}(t)$$  is treated as being normally distributed.


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

The approximate confidence bounds on the instantaneous failure intensity are then estimated from:


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


 * where


 * $${{\lambda }_{i}}(t)=\lambda \beta {{t}^{\beta -1}}$$


 * $$\begin{align}

& Var({{{\hat{\lambda }}}_{i}}(t))= & {{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & & +2\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda }) \end{align}$$

The variance calculation is the same as Eqn. (variance1) and:


 * $$\begin{align}

& \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}+\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}\ln t \\ & \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda }= & \hat{\beta }{{t}^{\hat{\beta }-1}} \end{align}$$

Crow Bounds
The Crow instantaneous failure intensity confidence bounds are given as:


 * $$\begin{align}

& {{\lambda }_{i}}{{(t)}_{L}}= & \frac{1} \\ & {{\lambda }_{i}}{{(t)}_{U}}= & \frac{1} \end{align}$$