On the Moments of the Trace of a Matrix and Approximations to Its Non- Central 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], [18] 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], [5], [18], [19] with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY'C'}$ 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 $\mathbf{Y'Y}$ 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 [6], [18]: \begin{equation*}\tag{1.1}(f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Now define $U^{(s)} = \mathrm{tr} (\mathbf{I}_p - \mathbf{YY}')^{-1} - p = \mathrm{tr} (\mathbf{I}_{f_2} - \mathbf{Y'Y}^{-1} - f_2$. In view of (1.1), we only consider $U^{(s)}$ when $s = p$, i.e. $U^{(p)}$, based on the density function [9] of $\mathbf{L} = \mathbf{YY}'$ for $f_2 \geqq p$. The first four moments of $U^{(s)}$ have been studied by Pillai in the central case [11], [12], [13], [14], [17] those for $U^{(2)}$ also by Pillai [15] in the non-central (linear) case and the first two moments of $U^{(p)}$ by the authors [7]. These results are extended in the present paper, obtaining the third and fourth moments of $U^{(p)}$ and further, two approximations to the distribution of $U^{(p)}$ are suggested in the linear case.