On the Non-central Distribution of the Second Elementary Symmetric Function of the Roots of a Matrix

Abstract

Let $\mathbf{X}$ be a $p \times f$ matrix variate $(p \leqq f)$ whose columns are independently normally distributed with $E(\mathbf{X}) = \mathbf{M}$ and covariance matrix $\mathbf{\Sigma}$. Let $w_1, \cdots, w_p$ be the characteristic roots of $|\mathbf{XX}' - w\mathbf{\Sigma}| = 0,$ then the distribution of $\mathbf{W} = \operatorname{diag} (w_i)$ is given by [4], [5] \begin{equation*}\tag{1.1} e^{-\frac{1}{2}\operatorname{tr}\mathbf{\Omega}}_0F_1(\frac{1}{2}f; \frac{1}{4}\mathbf{\Omega}, \mathbf{W})\kappa (p, f) \cdot e^{-\frac{1}{2}\operatorname{tr}\mathbf{W}}|\mathbf{W}|^{\frac{1}{2}(f-p-1)} \mathbf{\prod}_{i>j} (w_i - w_j), 0 < w_1 \leqq \cdots \leqq w_p < \infty,\end{equation*} where \begin{equation*}\tag{1.2}\kappa(p, f) = \pi^{\frac{1}{2}p^2}/\{2^{\frac{1}{2}pf}\Gamma_p(\frac{1}{2}f)\Gamma_p (\frac{1}{2}p)\},\end{equation*} $\mathbf{\Omega} = \operatorname{diag} (\omega_i)$ where $\omega_i, i = 1, \cdots, p,$ are the characteristic roots of $|\mathbf{MM}' - \mathbf{\omega\Sigma}| = 0$ and $_0F_1$ is the hypergeometric function of matrix argument (see Section 2) defined in [5]. The above distribution of non-central means with known covariance matrix was obtained by James [4]. But (1.1) has also been shown, [5], to be the limiting distribution as $n \rightarrow \infty$ of $n\mathbf{R}^2 = \mathbf{W}$ such that $0 < n\mathbf{P}^2 = \mathbf{\Omega} < \infty,$ where $\mathbf{R}^2 = \operatorname{diag} (r_i^2)$ and $\mathbf{P}^2 = operatorname{diag} (\rho^2_i)$ and where the canonical correlation coefficients $r_1^2, \cdots, r_p^2$ between a $p$-set and a $q$-set of variates $(p \leqq q)$ following a $(p + q)$ variate normal distribution, are calculated from a sample of $n + 1$ observations and $\rho_1^2, \cdots, \rho_p^2$ are population canonical correlation coefficients. Further $q = f.$ In this paper, the first two non-central moments of $W_2^{(p)},$ the second elementary symmetric function (esf) in $\frac{1}{2}w_1, \frac{1}{2}w_2,\cdots, \frac{1}{2}w_p$ have been obtained first by evaluating certain integrals involving zonal polynomials, and then alternately in terms of generalized Laguerre polynomials, [2], [5]. These moments were used to suggest an approximation to the non-central distribution of $W_2^{(p)}$. The approximation is observed to be good even for small values of $f.$