Template:Bounds on projected failure intensity rga

Fisher Matrix Bounds
The projected failure intensity $${{\lambda }_{P}}(T)$$  must be positive, thus  $$\ln {{\lambda }_{P}}(T)$$  is approximately treated as being normally distributed as well:


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


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

where:
 * •	 $${{\hat{\lambda }}_{P}}(T)=\tfrac{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{T}+\overline{d}\tfrac{M}{T}\bar{\beta }$$ when there are no BC modes.
 * •	 $${{\hat{\lambda }}_{P}}(T)={{\widehat{\lambda }}_{EM}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{T}+\overline{d}\widehat{h}(T|BD)$$ when there are BC modes.
 * •	 $${{N}_{i}}$$ is the total failure number of the  $${{i}^{th}}$$  distinct BD mode.

You can then get:


 * $$Var({{\lambda }_{P}}(T))\approx Var({{\hat{\gamma }}_{GP}})+\mu _{d}^{2}Var(h(T))\approx \frac{T}+\mu _{d}^{2}Var(h(T))$$


 * where:


 * $$\begin{align}

& \hat{h}(T)= & \frac{M}{T}\bar{\beta } \\ & Var(\hat{h}(T))= & {{(\frac{M}{T})}^{2}}Var(\bar{\beta })={{(\frac{M}{T})}^{2}}{{(\frac{M}{M-1})}^{2}}Var(\hat{\beta })=\fracVar(\hat{\beta }) \end{align}$$

The $$Var(\hat{\beta })$$  can be obtained from Fisher Matrix based on  $$M$$  distinct BD modes.

Crow Bounds

 * $$\begin{align}

& {{[{{\lambda }_{P}}(T)]}_{L}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{4}} \\ & {{[{{\lambda }_{P}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{4}} \end{align}$$

where $$C=\tfrac{\sqrt{T}}$$.