Chi Distribution

Let $ x_n=\vert g_n\vert$ where $ g_n$ follows the standard normal distribution $ N(0,1)$. Then $ x_n$ are Chi-distributed (Section 16.1.3)Let $ h(x)=x^2$. Then, for $ {\rm Re}(\lambda)<\frac{1}{2}$,

$\displaystyle \begin{array}{rcl}
e(u,\lambda)&=&{\displaystyle \int_u^\infty} ...
... && \;\;\;
\; \frac{1}{\sqrt{1-2\lambda}}
\mbox{ for } \; u<0.
\end{array}$

Also,

$\displaystyle \begin{array}{rcl}
c(u,\lambda)
&=& \frac{1-2\Phi(-u\sqrt{1-2\l...
...-2\lambda}}
\;\;\; \mbox{ for } \; u>0; \; 0 \mbox{ for } \; u<0.
\end{array}$

We compare the above results with the SPA in Section 7.2.



Baggenstoss 2017-05-19