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.
Citation
C. G. Khatri. K. C. S. Pillai. "On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices and Approximations to a Distribution." Ann. Math. Statist. 39 (4) 1274 - 1281, August, 1968. https://doi.org/10.1214/aoms/1177698252
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