Title: Characterizations of Noncentral Chi-Squared-Generating Covariance Structures for a Normally Distributed Random Vector

Author(s): Phil D. Young and Dean M. Young
Issue: Volume 78 Series A Part 2 Year 2016
Pages: 231 -- 247
Let ${\bf y} \sim N_n ({\bf \mu}, {\bf V})$, where ${\bf y}$ is a $n \times 1$ random vector and ${\bf V}$ is a $n \times n$ covariance matrix. We explicitly characterize the general form of the covariance structure ${\bf V}$ for which the family of quadratic forms $\{{\bf y}' {\bf A}_i {\bf y}\}_{i=1}^k$ for $i \in \{1, \ldots, k\}$, $2 \le k \le n$, is distributed as multiples of mutually independent non-central chi-squared random variables. We consider the case when the ${\bf A}_i$’s and ${\bf V}$ are both nonnegative definite, including several cases where the ${\bf A}_i$’s have special properties, and the case where the ${\bf A}_i$’s are symmetric and ${\bf V}$ is positive definite. Our results generalize the work of Pavur (Sankhyā ${\bf 51}$, 382–389, 1989), Baldessari (Comm. Statist. - Theory Meth. ${\bf 16}$, 785–803, 1987), and Chaganty and Vaish (Linear Algebra Appl. ${\bf 264}$, 421–437, 1997).
AMS (2000) subject classification. 15A24, 15A09, 62H10.
Keywords and phrases: Generalized inverse, Moore-Penrose pseudo-inverse, Non-central chi-squared random variable, Nonnegative-definite covariance matrix, Positive-definite covariance matrix.