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

Abstract

Let \({\mathbf {y}} \sim N_{n}\left ({\boldsymbol {\mu }}, {\mathbf {V}} \right )\), where y is a n×1 random vector and V is a n×n covariance matrix. We explicitly characterize the general form of the covariance structure V for which the family of quadratic forms \(\left \{{\mathbf {y}}^{\prime } {\mathbf {A}}_{i}{\mathbf {y}} \right \}^{k}_{i=1}\) for \(i \in \left \{1,...,k \right \}\), 2≤k≤n, is distributed as multiples of mutually independent non-central chi-squared random variables. We consider the case when the Ai’s and V are both nonnegative definite, including several cases where the Ai’s have special properties, and the case where the Ai’s are symmetric and V is positive definite. Our results generalize the work of Pavur (Sankhyā 51, 382–389, 1989), Baldessari (Comm. Statist. - Theory Meth. 16, 785–803, 1987), and Chaganty and Vaish (Linear Algebra Appl. 264, 421–437, 1997).