Distribution Theory of a Positive Definite Quadratic Form with Matrix Argument

Abstract

Let $x: 1 \times n$ be a row vector of random variables such that each $x_i (i = 1,2, \cdots, n)$ is independently normally distributed with mean $\mu_i$ and variance one. Many authors have studied the distribution of what we shall call a univariate quadratic form $xAx'$, where $A$ is a positive definite $n \times n$ matrix. Three types of representation of the distribution have been developed, namely (i) power series about the origin, (ii) mixtures of chi-squares (we refer to these as Ruben-type representations), and (iii) series of Laguerre polynomials. Though all three types of expansion yield correct convergent representations, it has been found that the Laguerre series representation is computationally the most convenient and effective throughout the range of interesting values of the argument. Let $X: p \times n$ be a matrix whose column vectors are independently and identically distributed in multivariate normal distributions having zero mean vector and variance covariance matrix $\Sigma$. If $L$ is a positive definite $n \times n$ matrix, we refer to the $p \times p$ matrix $S = XLX'$ as a positive definite quadratic form with matrix argument. Khatri [11] has given a representation of the density function of the distribution of $S$, somewhat similar to the Ruben-type expansion for the univariate case. In this paper we express the density function of $S$ in terms of Laguerre polynomials with matrix argument. Our results can be easily extended to quadratic forms in a matrix argument when the common multivariate distribution of the column vectors is complex. In Section 2, we give definitions and notations. Section 3 gives some results on integration over orthogonal groups, and in Section 4 we derive the main results.