Some Distribution Problems Connected with the Characteristic Roots of $S_1S^{-1}_2$

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 (ch) roots of $\mathbf{S}_1\mathbf{S}^{-1}_2$ and $\mathbf{\sigma}_1\mathbf{\sigma}^{-1}_2$ be denoted by $f_i (i = 1, 2,\cdots, p)$ and $\lambda_i(i = 1, 2, \cdots, p)$ respectively such that $0 < f_1 < f_2 < \cdots < f_p < \infty$ and $0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_p < \infty$. The distribution of $f_1, f_2, \cdots, f_P$ as stated by James [5] is not convenient for further development and is slowly convergent for higher values of $f_i's$. The distribution of $(f_1, \cdots, f_p)$ mentioned by James [5] can be written as \begin{equation*}\tag{1} c|\mathbf{\lambda}|^{-\frac{1}{2}n_1}|\mathbf{F}|^{\frac{1}{2}(n_1 - p - 1} \alpha_p(\mathbf{F} \int_{O(p)}|\mathbf{I}_p + \mathbf{\lambda}^{-1}\mathbf{HFH}'|^{-\frac{1}{2}(n_1 + n_2)} d\mathbf{H}\end{equation*} where \begin{equation*}\tag{2}c = \pi^{\frac{1}{2}p^2} \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)} \Pi^p_{j = 1} \Gamma (t - \frac{1}{2}j + \frac{1}{2},\end{equation*} \begin{equation*} {3}\alpha_p(\mathbf{F}) = \Pi ^{p - 1}_{i = 1} \Pi^p_{j = i + 1} (f_j - f_i), \quad \mathbf{F} = \operatorname{diag} (f_1, f_2, \cdots, f_p), \mathbf{\lambda} = \operatorname{diag} (\lambda_1, \lambda_2, \cdots, \lambda_p) \end{equation*} and the intergral is over an orthogonal group $O(p)$ with $\int_{O(p)} d\mathbf{H} = 1$. For testing the null hypothesis $H_0(\lambda\mathbf{\lambda} = \mathbf{I}_p), \lambda > 0$ being given, we have two statistics given by \begin{equation*}\tag{4}(i) l = |\lambda\mathbf{F}|^{n_1}/|\mathbf{I}_p + \lambda F|^{n_1 + n_2} \text{and} (ii) \lambda f_P \text{or} \lambda f_p/(1 + \lambda f_p)\end{equation*}. $l$ is considered by Anderson [1] and $\lambda f_p$ is obtained by Roy [7]. (1) is rewritten in such a way that the joint density function of $(\lambda f_1, \lambda f_2, \cdots, \lambda f_p)$ has noncentral parameters $\mathbf{I}_p - (\lambda\mathbf{\lambda})^{-1}$ and it is given by \begin{equation*}\tag{5}c|\lambda\mathbf{\lambda} |^{\frac{1}{2}n_1}|\lambda\mathbf{F}|^{\frac{1}{2}(n_1-p-1} \alpha_p(\lambda\mathbf{F})| \mathbf{I}_p + \lambda\mathbf{F})| ^{-\frac{1}{2}(n_1 + n_2)} _1F_0^{(p)} (\frac{1}{2}n_1 + \frac{1}{2}n_2; \mathbf{I}_p - (\lambda\mathbf{\lambda}^{-1}, \lambda\mathbf{F}(\mathbf{I}_p + \lambda\mathbf{F})^{-1})\end{equation*}. Hence, similar to testing of means, we propose the statistics $T = tr (\lambda\mathbf{F})$ and $V = tr (\lambda\mathbf{F})(\mathbf{I}_p + \lambda\mathbf{F})^{-1}$ for testing the hypothesis $H_0$ and obtain their distribution of $T$ only while the moment generating function of $V$ is given. Moreover, if in the null hypothesis $H'_0(\lambda\mathbf{\lambda} = \mathbf{I}_p, \lambda > 0$ is unknown, then the test procedure will depend on the ratios of roots, and hence, we consider the joint distribution of $(x_1, x_2, \cdots, x_{p - 1})$, where $x_i = f_i/f_p$ for $i = 1, 2, \cdots, p - 1$, and $f_p$. Under null hypothesis $H'_0$, we obtain the density functions of $x_1$ (or $x_{p - 1}$) and of $(y_2, y_3, \cdots, y_{p - 1})$ for $y_i = (f_i - f_1)/f_p - f_1), i = 2, 3, \cdots, p - 1$.