Template:Bounds on growth potential failure intensity rga

Fisher Matrix Bounds
If there are no BC failure modes, the growth potential failure intensity is $${{\widehat{r}}_{GP}}(T)=\tfrac{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{T}$$.


 * Then:


 * $$\begin{align}

& Var({{\widehat{r}}_{GP}})= & \frac{1}{T}\left[ \frac{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,{{(1-{{d}_{i}})}^{2}}\frac{T} \right] \\ & \le & \frac{1}{T}\left[ \frac{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T} \right] \\ & = & \frac{T} \end{align}$$

If there are BC failure modes, the growth potential failure intensity is $${{\widehat{r}}_{GP}}(T)={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{T},$$   $$Var({{\widehat{r}}_{GP}})\approx \tfrac{T}$$. Therefore:


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

The confidence bounds on the growth potential failure intensity are as follows:


 * $$\begin{align}

& {{r}_{L}}= & {{{\hat{r}}}_{GP}}+\frac{2}-\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{4}} \\ & {{r}_{U}}= & {{{\hat{r}}}_{GP}}+\frac{2}+\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{4}} \end{align}$$

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

Crow Bounds
The Crow bounds for the growth potential failure intensity are the same as the Fisher Matrix bounds.