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

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

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.


Baggenstoss 2017-05-19