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

Abstract

Let $X:p imes n$ 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 $Y:p imes m$ 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 egin{equation*} ag{1}_pF_q(a_1, cdots, a_p; b_1, cdots, b_q; Z) end{equation*} $= sum^infty_{k = 0} sum_kappa lbrack (a_1)_kappa cdots (a_p)_kappa/(b_1)_kappa cdots (b_q)_kappa bracklbrack C_kappa(Z)/k! brack$ 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), egin{equation*} ag{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 egin{equation*} ag{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 $m imes m$ 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 egin{equation*} ag{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_kappalbrack (a_1)_kappa cdots (a_p)_kappa/(b_1)_kappa cdots (b_q)_kappa bracklbrack C_kappa(S)C_kappa(R)/C_kappa(I_m)k! brack.$ 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.