Saddlepoint Approximation for Linear Function of Real Positive Data

Let ${\bf x}$ be a set of $N$ independent chi-squared RV. Let $x_i$ be chi-squared with $k_i$ degrees of freedom, $1\leq i \leq N$. Let ${\bf A}$ be an $N$-by-$P$ full-rank matrix and let ${\bf z}$ be the $P\times 1$ feature vector

$${\bf z}={\bf A}^\prime {\bf x}.$$

Note that for the standard chi-squared distributions, the expected value of $z_i$ equals $k_i$. We have previously published the SPA for the PDF $p({\bf z})$ in Kay, Nuttall, and Baggenstoss [16]. A short summary of these results is given below.