On an Asymptotic Representation of the Distribution of the Characteristic Roots of $S_1S_2^{-1}$

Abstract

Let $\mathbf{S}_i: p \times p(i = 1, 2)$ be independently distributed as Wishart $(n_i, p, \mathbf{\Sigma}_i)$. Let the characteristic roots of $\mathbf{S}_1 \mathbf{S}_2^{-1}$ and $\mathbf{\Sigma}_1 \mathbf{\Sigma}_2^{-1}$ be denoted by $l_i (i = 1,2, \cdots, p)$ and $\lambda_i (i = 1,2, \cdots, p)$ respectively such that $l_1 > l_2 > \cdots > l_p > 0$ and $\lambda_1 > \lambda_2 > \cdots > \lambda_p > 0$. Then the distribution of $l_1, \cdots, l_p$ can be expressed in the form (Khatri [8]) \begin{equation*} \tag{1.1} C|\mathbf{\Lambda}|^{-\frac{1}{2}n_1}|\mathbf{L}|^{\frac{1}{2}(n_1-p- 1)}\{\prod^p_{i<j}(l_j - l_i)\} \int_{O(p)}|\mathbf{I}_p + \mathbf{\Lambda}^{-1}\mathbf{HLH}'|^{-\frac{1}{2}(n_1+n_2)}(\mathbf{H}' d\mathbf{H})\end{equation*} where \begin{equation*}\begin{split}C = 2^{-p}\pi^{\frac{1}{4} p(p-1)}\{\prod^p_{i=1} \mathbf{\Gamma}(i/2)\} \mathbf{\Gamma}_p(\frac{1}{2}n_1 + frac{1}{2}n_2)\{\Gamma_ p(\frac{1}{2}p)\Gamma_p(\frac{1}{2}n_1)\Gamma_ p(\frac{1}{2}n_2)\}^{-1}, \\ \Gamma_p(t) = \pi^{\frac{1}{4}p(p-1)} \prod^p_{j=1} \Gamma(t - \frac{1}{2}j + \frac{1}{2}), \mathbf{L} = \operatorname{diag}(l_1, \cdots, l_p), \mathbf{\Lambda} = \operatorname{diag}(\lambda_1, \cdots, \lambda_p)\end{split}\end{equation*} and $(\mathbf{H}' d\mathbf{H})$ is the invariant measure on the group $O(p)$. However, this form is not convenient for further development. Also, since \begin{align*} \tag{1.2} I &= \int_{O(p)}|\mathbf{I}_p + \mathbf{\Lambda}^{-1} \mathbf{HLH}'|^{\frac{1}{2}(n_1 + n_2)}(\mathbf{H}' d\mathbf{H}) \\ &= C' \sum^\infty_{k=0} \frac{1}{k!} \sum_\kappa\frac{C_\kappa(-\mathbf{\Lambda}^{-1})C_\kappa(\mathbf{L})(n_1 + n_2)_\kappa}{C_\kappa(\mathbf{I}_p)}\end{align*} where $C' = 2^p\pi^{\frac{1}{4}p(p+1)}/\prod^p_{i=1} \Gamma(i/2)$ and the zonal polynomial $C_\kappa(\mathbf{T})$ of any $p \times p$ symmetric matrix $\mathbf{T}$ is defined in James [7], where $k$ is a partition of $k$ into not more than $p$ parts, the use of (1.2) in (1.1) gives a power series expansion, but the convergence of this series is very slow. In the one sample case G. A. Anderson [1] has obtained a gamma-type asymptotic expansion for the distribution of the characteristic roots of the estimated covariance matrix. In this paper we obtain a beta-type asymptotic representation of the roots distribution of $\mathbf{S}_1 \mathbf{S}_2^{-1}$ involving linkage factors between sample roots and corresponding population roots. If the roots are distinct the limiting distribution as $n_2$ tends to infinity has the same form as that of Anderson [1]. If, moreover, $n_1$ is assumed also large, then it agrees with Girshick's result [4], which was also discussed in Anderson [1].