Template:Bounds on growth potential failure intensity rga

Bounds on Growth Potential Failure Intensity

Fisher Matrix Bounds

If there are no BC failure modes, the growth potential failure intensity is [math]\displaystyle{ {{\widehat{r}}_{GP}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T} }[/math] .

Then:

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

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

[math]\displaystyle{ \sqrt{T}\left( \frac{{{{\hat{r}}}_{GP}}-{{r}_{GP}}}{\sqrt{{{r}_{GP}}}} \right)\sim N(0,1) }[/math]

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

[math]\displaystyle{ \begin{align} & {{r}_{L}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\ & {{r}_{U}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}\,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \end{align} }[/math]

where [math]\displaystyle{ C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}} }[/math] .

Crow Bounds

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