Open Access
Translator Disclaimer
April, 1971 Some Distribution Problems in the Multivariate Complex Gaussian Case
K. C. S. Pillai, G. M. Jouris
Ann. Math. Statist. 42(2): 517-525 (April, 1971). DOI: 10.1214/aoms/1177693402


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.


Download Citation

K. C. S. Pillai. G. M. Jouris. "Some Distribution Problems in the Multivariate Complex Gaussian Case." Ann. Math. Statist. 42 (2) 517 - 525, April, 1971.


Published: April, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0218.62050
MathSciNet: MR288912
Digital Object Identifier: 10.1214/aoms/1177693402

Rights: Copyright © 1971 Institute of Mathematical Statistics


Vol.42 • No. 2 • April, 1971
Back to Top