Template:Bounds on demonstrated failure intensity rga

Fisher Matrix Bounds
If there are no BC failure modes, the demonstrated failure intensity is $${{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{BD}}}{T}$$. Thus:


 * $$Var({{\hat{\lambda }}_{D}}(t))=\frac+\frac=\frac{{{\lambda }_{D}}(t)}{T}$$


 * and:


 * $$\sqrt{T}\left( \frac{{{{\hat{\lambda }}}_{D}}(T)-{{\lambda }_{D}}(T)}{\sqrt{{{\lambda }_{D}}(T)}} \right)\sim N(0,1)$$


 * $${{\lambda }_{D}}(T)={{\hat{\lambda }}_{D}}(T)+\frac{2}\pm \sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{4}}$$

where $$C=\tfrac{\sqrt{T}}$$. If there are BC failure modes, the demonstrated failure intensity, $${{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}$$, is actually the instantaneous failure intensity based on all of the data. $${{\lambda }_{CA}}(T)$$ must be positive, thus  $$\ln {{\lambda }_{CA}}(T)$$  is approximately treated as being normally distributed.


 * $$\frac{\ln {{{\hat{\lambda }}}_{CA}}(T)-\ln {{\lambda }_{CA}}(T)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{CA}}(T)})}\sim N(0,1)$$

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


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

where $${{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}$$.


 * $$\begin{align}

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

The variance calculation is the same as described in Chapter 5.

Crow Bounds
If there are no BC failure modes then:


 * $$\begin{align}

& {{[{{\lambda }_{D}}(T)]}_{l}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\ & {{[{{\lambda }_{D}}(T)]}_{u}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N} \end{align}$$

where $${{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}$$. If there are BC modes then the confidence bounds on the demonstrated failure intensity are calculated as presented in Chapter 5.