On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors

Abstract

Let $X:p \times n$ be a matrix of random real variates such that the column vectors of $X$ are independently and identically distributed as multivariate normals with zero mean vectors. Then a positive definite quadratic function in normal vectors is defined as $XLX$' where $L$ is a symmetric positive definite (p.d.) matrix with real elements. In the analysis of variance, such functions appear. In the previous study, Khatri [14], [16], has established the necessary and sufficient conditions for the independence and the Wishartness of such functions. In this paper, we study the distribution of a positive definite quadratic function and the distribution of $Y' (XLX')^{-1}Y$ where $Y:p \times m$ is independently distributed of $X$ and its columns are independently and identically distributed as multivariate normals with zero mean vectors. Moreover, we study the distribution of the characteristic (ch.) roots of $(YY')(XLX')^{-1}$ and the similar related problems. When $p = 1$, the distribution of a p.d. quadratic function in normal variates (central or noncentral) has been studied by a number of people (see references). In the study of the above and related topics in multivariate distribution theory, we are using zonal polynomials. A. T. James [10], [11], [12], [13], and Constantine [1], [2], have used them successfully and have given the final results in a very compact form, using hypergeometric functions $_pF_q(S)$ in matrix arguments. These functions are defined by \begin{equation*}\tag{1}_pF_q(a_1, \cdots, a_p; b_1, \cdots, b_q; Z) \end{equation*} $= \sum^\infty_{k = 0} \sum_\kappa \lbrack (a_1)_\kappa \cdots (a_p)_\kappa/(b_1)_\kappa \cdots (b_q)_\kappa\rbrack\lbrack C_\kappa(Z)/k!\rbrack$ where $C_\kappa(Z)$ is a symmetric homogeneous polynomial of degree $k$ in the latent roots of $Z$, called zonal polynomials (for more detail study of zonal polynomials, see the references of A. T. James and Constantine), $\kappa = (k_1, \cdots, k_p), k_1 \geqq k_2 \geqq \cdots \geqq k_p \geqq 0, k_1 + k_2 + \cdots + k_p = k; a_1, \cdots, a_p, b_1, \cdots, b_q$ are real or complex constants, none of the $b_j$ is an integer or half integer $\leqq \frac{1}{2}(m - 1)$ (otherwise some of the denominators in (1) will vanish), \begin{equation*}\tag{2}(a)_\kappa = \prod^m_{j = 1} (a - \frac{1}{2}(j - 1))_{kj} = \Gamma_m(a,\kappa)/\Gamma_m(a), \end{equation*} (x)_n = x(x + 1) \cdots (x + n - 1), (x)_0 = 1$ and \begin{equation*}\tag{3}\Gamma_m(a) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a - \frac{1}{2}(j - 1)) \end{equation*} and $\Gamma_m(a, \kappa) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a + k_j - \frac{1}{2}(j - 1)).$$ In (1), $Z$ is a complex symmetric $m \times m$ matrix, and it is assumed that $p \leqq q + 1$, otherwise the series may converge for $Z = 0$. For $p = q + 1$, the series converge for $\|Z\| < 1$, where $\|Z\|$ denote the maximum of the absolute value of ch. roots of $Z$. For $p \leqq q$, the series converge for all $Z$. Similarly we define \begin{equation*}\tag{2b}_pF^{(m)}_q (a_1, a_2, \cdots, a_p; b_1, \cdots, b_q; S, R)\end{equation*} $ = \sum^\infty_{k = 0} \sum_\kappa\lbrack (a_1)_\kappa \cdots (a_p)_\kappa/(b_1)_\kappa \cdots (b_q)_\kappa\rbrack\lbrack C_\kappa(S)C_\kappa(R)/C_\kappa(I_m)k!\rbrack.$ The Section 2 gives some results on integration with the help of zonal polynomials, the Section 3 derives the distributions based on p.d. quadratic functions, the Section 4 gives the moments of certain statistics arising in the study of multivariate distributions, and the Section 5 gives the results for complex multivariate Gaussian variates.