Template:Bounds on instantaneous failure intensity rsa

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


 * $$\frac{\ln ({{\widehat{\lambda }}_{i}}(t))-\ln ({{\lambda }_{i}}(t))}{\sqrt{Var\left[ \ln ({{\widehat{\lambda }}_{i}}(t)) \right]}}\sim N(0,1)$$

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


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

where $${{\lambda }_{i}}(t)=\lambda \beta {{t}^{\beta -1}}$$  and:


 * $$\begin{align}

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

The variance calculation is the same as Eqns. (var1), (var2) and (var3):


 * $$\begin{align}

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