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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.

Let $\mathbf{X}$ be a $p \times f_2$ matrix variate $(p \leqq f_2)$ and $\mathbf{Y}$ a $p \times f_1$ matrix variate $(p \leqq f_1)$ and the columns be all independently normally distributed with covariance matrix $\mathbf{\Sigma}, E(\mathbf({X}) = \mathbf{M}$ and $E(\mathbf{Y}) = \mathbf{0}$. Let $0 < l_1 \leqq \cdots \leqq l_p < 1$ be the ordered characteristic roots of \begin{equation*}\tag{1.1} |\mathbf{XX}' - l(\mathbf{YY}' + \mathbf{XX}')| = \mathbf{0}\end{equation*} and $\omega_1, \cdots, \omega_p$ those of $|\mathbf{MM}' - \omega\mathbf{\Sigma}| = \mathbf{0},$ then the joint density function of $l_1, \cdots, l_p$ is given by Constantine [1], James [2] in the form \begin{equation*}\tag{1.2}\exp (-\frac{1}{2} \operatorname{tr} \mathbf{\Omega}) _1F_1(\frac{1}{2}\nu; \frac{1}{2}f_2; \frac{1}{2}\mathbf{\Omega}, \mathbf{L})f_0(l_1, \cdots, l_p),\end{equation*} where \begin{equation*}\begin{align*}{1.3}f_0(l_1, \cdots, l_p) = C(p, f_1, f_2) \mathbf{\prod}^p_{i=1} \{l_1^{\frac{1}{2}(f_2-p-1)}(1 - l_i)^{\frac{1}{2}(f_1-p-1)}\}\alpha_p (\mathbf{L}), \\ \mathbf{L} = \mathbf{X}'(\mathbf{YY}' + \mathbf{XX}')^{-1}\mathbf{X},\quad \mathbf{\Omega} = \mathbf{M}'\mathbf{\Sigma}^{-1}\mathbf{M},\quad \nu = f_1 + f_2\end{align*},\end{equation*} $_1F_1$ is the hypergeometric function of matrix argument defined in [2] given by $\sum^\infty_{k=0} \sum_\mathbf{kappa}{2}\nu)_\mathbf{kappa}C_\mathbf{kappa}(\frac{1}{2} \mathbf{\Omega})_\mathbf{kappa}(\mathbf{L})/(\frac{1}{2}f_2)\mathbf{kappa} C_\mathbf{kappa}(\mathbf{I}_p)k !$ and where $C(p, f_1, f_2) = \pi^{\frac{1}{2}p^2}\Gamma_p(\frac{1}{2}\nu)/\{\Gamma_p(\frac{1}{2} f_1)\Gamma_p(\frac{1}{2}f_2)\Gamma_p(\frac{1}{2}p)\}$ and $\alpha_p(\mathbf{L} = \mathbf{\prod}_{i>j}(l_i - l_j).$ In this paper the distribution of Pillai's $V^{(p)}$ criterion which is the trace of $\mathbf{L}$, [5], [6] and that of Roy's largest root criterion, $l_p$, [8], [10], have been obtained in series forms and certain constants involved in the series tabulated. In addition, the first four moments of $V^{(p)}$ are also obtained in the linear case, illustrating further use of some of the tabulations.

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 $y_i = (f_i - f_1)/f_p - f_1), i = 2, 3, \cdots, p - 1$.

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order of $p, \mathbf{A}_1$, positive definite and having a Wishart distribution [2], [12] 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], [4], [12], [13]) with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY'C'}$ where $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'$ and the density function of $\mathbf{Y} : p \times f_2$ is given by \begin{equation*}\tag{1.1}k_1e^{-\lambda^2} \sum^\infty_{j = 0} (2\lambda y_{11})^j\Gamma\lbrack\frac{1}{2}(f_1 + f_2 + j)\rbrack |\mathbf{I}_p - \mathbf{YY}'|^{\frac{1}{2}}(f_{1 - p - 1)}/j!\end{equation*} where $\mathbf{I}_p$ is an identity matrix of order $p$, $k_1 = \prod^p_{i = 2} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 - i + 1)\rbrack/ \pi^{\frac{1}{2}pf_2} \prod^p_{i = 1} \Gamma\lbrack f_1 - i = 1)/2\rbrack,$ $\lambda$ is the only non-centrality parameter in the linear case and $y_{11}$ is the element in the top left corner of the $\mathbf{Y}$ matrix. Now $V^{(s)}$ criterion suggested by Pillai and $U^{(s)}$ (a constant times Hotelling's $T_0^2$), [7], [8], [9], [10] are the sums of the non-zero characteristic roots of the matrix $\mathbf{YY}'$ and $(\mathbf{I}_p - \mathbf{YY}')^{-1} - \mathbf{I}_p$ respectively. Here $s$ is minimum $(f_2, p)$. Also we may note that $V^{(s)} = \text{trace} \mathbf{YY}' = \text{trace} \mathbf{Y'Y}$ and $U^{(s)} = \mathrm{tr} (\mathbf{I}_p - \mathbf{YY}')^{-1} - p = \mathrm{tr} (\mathbf{I}_{f_2} - \mathbf{Y'Y})^{-1} - f_2$. It can be shown that the density function of the characteristic roots of the matrix $\mathbf{Y'Y}$ 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: ([12], [5]) \begin{equation*}\tag{1.2}(f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Hence, for the criterion $V^{(s)}$, (and similarly for $U^{(s)}$), we shall only consider the density function of $\mathbf{L} = \mathbf{YY}'$ for $f_2 \geqq p$ which is given by [6] \begin{equation*}\tag{1.3}f(\mathbf{L}) = ke^{-\lambda^2}_1F_1\{\frac{1}{2} (f_1 + f_2), \frac{1}{2}f_2, \lambda^2l_{11}\}|\mathbf{L}|^{ (f_2 - p - 1)/2} |\mathbf{I}_p - \mathbf{L}|^{(f_1 - p - 1)/2}, \end{equation*} where $k = \pi^{-p(p - 1)/4} \prod^p_{i = 1} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 + 1 - i)\rbrack/\{\Gamma\lbrack \frac{1}{2}(f_1 + 1 - i)\rbrack\Gamma\lbrack\frac{1}{2}(f_2 + 1 - i)\rbrack\}.$ $l_{11}$ is the element in the top left corner of the matrix $\mathbf{L}$ and $_1F_1$ denotes the confluent hypergeometric function. We shall call the distribution of $\mathbf{L}: p \times p$ the non-central (linear) multivariate beta distribution with $f_2$ and $f_1$ degrees of freedom. Pillai [11] had noted that the elements of the matrix $\mathbf{L}$ can be transformed into independent beta variables which he showed for $p = 2, 3, 4$ and $5$. In this paper we give a theorem which proves the general case. In addition, when $\lambda = 0$ the first and second order moments of $l_{ij}$ are obtained and used to derive the first two moments of $V^{(s)}$ in the non-central case when $f_2 \geqq p$. The moments of $V^{(s)}$ for $f_2 \leqq p$ can be written down with the help of (1.2). Similar results are obtained for $U^{(s)}$.

In this paper a decision procedure based on the Tukey Studentized range ([5], [6], [8]) has been shown to be an optimum procedure for a particular type of slippage of means of univariate normal populations based on a common but unknown variance. The method given here is similar to that used by Paulson [2] and Truax [7].