Some Results on the Non-Central Multivariate Beta Distribution and Moments of Traces of Two Matrices

Abstract

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order of $p, \mathbf{A}_1$, positive definite and having a Wishart distribution [2], [12] with $f_1$ degrees of freedom, and $\mathbf{A}_2$, at least positive semi-definite and having a (pseudo) non-central (linear) Wishart distribution ([1], [3], [4], [12], [13]) with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY'C'}$ where $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'$ and the density function of $\mathbf{Y} : p \times f_2$ is given by \begin{equation*}\tag{1.1}k_1e^{-\lambda^2} \sum^\infty_{j = 0} (2\lambda y_{11})^j\Gamma\lbrack\frac{1}{2}(f_1 + f_2 + j)\rbrack |\mathbf{I}_p - \mathbf{YY}'|^{\frac{1}{2}}(f_{1 - p - 1)}/j!\end{equation*} where $\mathbf{I}_p$ is an identity matrix of order $p$, $k_1 = \prod^p_{i = 2} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 - i + 1)\rbrack/ \pi^{\frac{1}{2}pf_2} \prod^p_{i = 1} \Gamma\lbrack f_1 - i = 1)/2\rbrack,$ $\lambda$ is the only non-centrality parameter in the linear case and $y_{11}$ is the element in the top left corner of the $\mathbf{Y}$ matrix. Now $V^{(s)}$ criterion suggested by Pillai and $U^{(s)}$ (a constant times Hotelling's $T_0^2$), [7], [8], [9], [10] are the sums of the non-zero characteristic roots of the matrix $\mathbf{YY}'$ and $(\mathbf{I}_p - \mathbf{YY}')^{-1} - \mathbf{I}_p$ respectively. Here $s$ is minimum $(f_2, p)$. Also we may note that $V^{(s)} = \text{trace} \mathbf{YY}' = \text{trace} \mathbf{Y'Y}$ and $U^{(s)} = \mathrm{tr} (\mathbf{I}_p - \mathbf{YY}')^{-1} - p = \mathrm{tr} (\mathbf{I}_{f_2} - \mathbf{Y'Y})^{-1} - f_2$. It can be shown that the density function of the characteristic roots of the matrix $\mathbf{Y'Y}$ for $f_2 \leqq p$ can be obtained from that of the characteristic roots of $\mathbf{YY}'$ for $f_2 \geqq p$ if in the latter case the following changes are made: ([12], [5]) \begin{equation*}\tag{1.2}(f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Hence, for the criterion $V^{(s)}$, (and similarly for $U^{(s)}$), we shall only consider the density function of $\mathbf{L} = \mathbf{YY}'$ for $f_2 \geqq p$ which is given by [6] \begin{equation*}\tag{1.3}f(\mathbf{L}) = ke^{-\lambda^2}_1F_1\{\frac{1}{2} (f_1 + f_2), \frac{1}{2}f_2, \lambda^2l_{11}\}|\mathbf{L}|^{ (f_2 - p - 1)/2} |\mathbf{I}_p - \mathbf{L}|^{(f_1 - p - 1)/2}, \end{equation*} where $k = \pi^{-p(p - 1)/4} \prod^p_{i = 1} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 + 1 - i)\rbrack/\{\Gamma\lbrack \frac{1}{2}(f_1 + 1 - i)\rbrack\Gamma\lbrack\frac{1}{2}(f_2 + 1 - i)\rbrack\}.$ $l_{11}$ is the element in the top left corner of the matrix $\mathbf{L}$ and $_1F_1$ denotes the confluent hypergeometric function. We shall call the distribution of $\mathbf{L}: p \times p$ the non-central (linear) multivariate beta distribution with $f_2$ and $f_1$ degrees of freedom. Pillai [11] had noted that the elements of the matrix $\mathbf{L}$ can be transformed into independent beta variables which he showed for $p = 2, 3, 4$ and $5$. In this paper we give a theorem which proves the general case. In addition, when $\lambda = 0$ the first and second order moments of $l_{ij}$ are obtained and used to derive the first two moments of $V^{(s)}$ in the non-central case when $f_2 \geqq p$. The moments of $V^{(s)}$ for $f_2 \leqq p$ can be written down with the help of (1.2). Similar results are obtained for $U^{(s)}$.