On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices and Approximations to a Distribution

Abstract

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order $p, \mathbf{A}_1$, positive definite and having a Wishart distribution ([2], [23]) 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], [23], [24]) with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY}'\mathbf{C}'$ where $\mathbf{Y}$ is $p \times f_2$ and $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'.$ Now consider the $s$( = minimum $(f_2, p))$ non-zero characteristic roots of the matrix $\mathbf{YY}'$. It can be shown that the density function of the characteristic roots of $\mathbf{Y}'\mathbf{Y}$ 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: [23] \begin{equation*}\tag{1.1} (f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Now, in view of (1.1), we consider only the case $s = p$, based on the density function [12] of $L = \mathbf{YY}'$ for $f_2 \geqq p$. In this paper, some results are obtained first regarding the $i$th elementary symmetric function (esf) of the characteristic roots of a non-singular matrix $\mathbf{P} (\operatorname{tr}_i\mathbf{P})$ which are useful to compute the moments of $\operatorname{tr}_i\mathbf{L}$ and $\operatorname{tr}_i\{(\mathbf{I} - \mathbf{L})^{-1} - \mathbf{I}\}$. In particular, the first two moments of $\operatorname{tr}_2 \mathbf{L}$ are obtained in the non-central linear case. These two moments of the above criteria in the central case have been obtained earlier by Pillai ([18], [19]). Further, from a study of the first four moments of $U^{(p)} = \operatorname{tr}\{(\mathbf{I} - \mathbf{L})^{-1} - \mathbf{I}\}$, [11], [14], two approximations to the distribution of $U^{(p)}$ were obtained in the general non-central case. The approximations are generalizations of those given by Khatri and Pillai [10] for the linear case. The accuracy comparisons of the approximations are also made.