Some Distribution Problems in the Multivariate Complex Gaussian Case

Abstract

Let $\mathbf{X}_1: p \times n$ and $\mathbf{X}_2: p \times n$ be real random variables having the joint density function \begin{equation*}\tag{1.1} (2\pi)^{-pn}|\mathbf{\Sigma}_0|^{-\frac{1}{2}n} \exp \{-\frac{1}{2} \operatorname{tr}\mathbf{\Sigma}_0^{-1}(\mathbf{X} - \mathbf{\nu})(\mathbf{X} - \mathbf{\nu})'\},\quad - \infty \leqq \mathbf{X} \leqq \infty\end{equation*} where \begin{equation*}\mathbf{X} = \binom{\mathbf{X}_1}{\mathbf{X}_2},\quad \mathbf{\Sigma}_o = \begin{pmatrix}\mathbf{\Sigma}_1 & -\mathbf{\Sigma}_2 \\ \mathbf{\Sigma}_2 & \mathbf{\Sigma}_1\end{pmatrix},\quad \nu = \begin{pmatrix}\mathbf{\mu}_1 & -\mathbf{\mu}_2 \\ \mathbf{\mu}_2 & \mathbf{\mu}_1\end{pmatrix} \binom{\mathbf{M}_1}{\mathbf{M}_2},\end{equation*} $\mathbf{\Sigma}_1: p \times p$ is a real symmetric positive definite (pd) matrix, $\mathbf{\Sigma}_2: p \times p$ is a real skew-symmetric matrix, $\mathbf{\mu}_j: p \times q$ and $\mathbf{M}_j: q \times n (j = 1, 2)$, are given matrices or their joint density does not contain $\mathbf{\Sigma}_1, \mathbf{\Sigma}_2, \mathbf{\mu}_1, \mathbf{\mu}_2$ as parameters. Then it has been shown by Goodman [10] that the distribution of the complex matrix $\mathbf{Z} = \mathbf{X}_1 + i\mathbf{X}_2, (i = (-1)^{\frac{1}{2}})$, is complex Gaussian and its density function is given by \begin{equation*}\tag{1.2} N_c(\mathbf{\mu}\mathbf{M}, \mathbf{\Sigma}) = \pi^{-pn}|\Sigma|^{-n} \exp \{ -\operatorname{tr} \mathbf{\Sigma}^{-1} (\mathbf{Z} - \mathbf{\mu M})(\mathbf{\bar{Z}} - \mathbf{\overline{\mu M}})'\}\end{equation*} where $\mathbf{\Sigma} = \mathbf{\Sigma}_1 + i\mathbf{\Sigma}_2$ is Hermitian pd, i.e. $\mathbf{\bar{\Sigma}}' = \mathbf{\Sigma, \mu} = \mathbf{\mu}_1 + i\mathbf{\mu}_2$ and $\mathbf{M} = \mathbf{M}_1 + i\mathbf{M}_2$. Goodman [5], Wooding [17], James [6], Al-Ani [1], and Khatri [8], [9], [10], [11] have studied distributions derived from a sample of a complex $p$-variate normal distribution. Some important concepts and necessary notation are given below. \begin{align*} \tilde\Gamma_m(a) &= \pi^{\frac{1}{2}m(m - 1)} \prod^m_{i = 1} \Gamma(a - i + 1), \\ \lbrack a \rbrack_\kappa &= \prod^m_{i = 1} (a - i + 1)_{k_i} = \tilde\Gamma_m(a, k)/\tilde\Gamma_m(a)\end{align*} where $\kappa = (k_1, k_2, \cdots, k_p)$ is a partition of the integer $k$ and $\tilde\Gamma_m(a, \kappa) = \pi^{\frac{1}{2}m(m - 1)} \prod^m_{i = 1} \Gamma(a + k_i - i + 1).$ The hypergeometric functions are defined as $_p\tilde{F}_q(a_1, \cdots, a_p; b_1, \cdots, b_q; \mathbf{A, B}) = \sum^\infty_{k = 0} \sum_k \frac{\prod^p_{i = 1} \lbrack a_i \rbrack_\kappa \tilde{C}_\kappa (A) \tilde{C}_\kappa (\mathbf{B})}{\prod^q_{i = 1} \lbrack b_i \rbrack_\kappa \tilde{C}_\kappa (\mathbf{I}_m)k!}$ or when $\mathbf{B} = \mathbf{I}_m$ we denote it by $_p\tilde{F}_q(a_1, \cdots, a_p; b_1, \cdots, b_q; \mathbf{A})$ and $\tilde{C}_\kappa(\mathbf{A})$ is a zonal polynomial of a Hermitian matrix $\mathbf{A}$ and is given as a symmetric function of the characteristic roots of $\mathbf{A}$. The non-central distributions of the characteristic roots concerning the classical problems of the equality of two covariance matrices, MANOVA model, and canonical correlation coefficients have been found by James [6] and Khatri [8], [10]. Here for the three cases mentioned, we give the general moment and the density which is expressed in terms of Meijer's $G$-function [13], [14], for $W^{(p)} = \prod^p_{i = 1} (1 - w_i)$, where the $w_i, i = 1, 2, \cdots, p$ are the characteristic roots in the above cases. The moments and densities are analogous to those given in the real case [7], [15]. Further the density functions of $U$ and Pillai's $V$ criteria in the complex central case are obtained for $p = 2$, and from the non-central complex multivariate $F$ distribution various independence relationships are shown and independent beta variables are obtained.