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The first four moments of the sum of $s$ non-null latent roots of a matrix occurring in multivariate analysis are studied. In particular, the first four moments of the sum of six roots are found, and are used to compare the upper percentage points obtained directly from the moment ratios with those from Pillai's approximate distribution.

New methods are introduced for deriving the sampling distributions of statistics obtained from a normal multivariate population. Exterior differential forms are used to represent the invariant measures on the orthogonal group and the Grassmann and Stiefel manifolds. The first part is devoted to a mathematical exposition of these. In the second part, the theory is applied; first, to the derivation of the distribution of the canonical correlation coefficients when the corresponding population parameters are zero; and secondly, to split the distribution of a normal multivariate sample into three independent distributions, (a) essentially the Wishart distribution, (b) the invariant distribution of a random plane which is given by the invariant measure on the Grassmann manifold, (c) the invariant distribution of a random orthogonal matrix. This decomposition provides derivations of the Wishart distribution and of the distribution of the latent roots of the sample variance covariance matrix when the population roots are equal.

As has been noted by several authors, when a multivariate normal distribution with correlation matrix $\{\rho_{ij}\}$ has a correlation structure of the form $\rho_{ij} = \alpha_i \alpha_j (i \neq j)$, where $-1 \leqq \alpha_i \leqq + 1$, its c.d.f. can be expressed as a single integral having a product of univariate normal c.d.f.'s in the integrand. The advantage of such a single integral representation is that it is easy to evaluate numerically. In this paper it is noted that the $n$-variate normal c.d.f. with correlation matrix $\{\rho_{ij}\}$ can always be written as a single integral in two ways, with an $n$-variate normal c.d.f. in the integrand and the integration extending over a doubly-infinite range, and with an $(n - 1)$-variate normal c.d.f. in the integrand and the integration extending over a singly-infinite range. We shall show that, for certain correlation structures, the multivariate normal c.d.f. in the integrand factorizes into a product of lower-order normal c.d.f.'s. The results may be useful in instances where these lower-order integrals are tabulated or can be evaluated. One important special case is $\rho_{ij} = \alpha_i \alpha_j$, previously mentioned. Another is $\rho_{ij} = \gamma_i/\gamma_j (i < j)$, where $|\gamma_i| < |\gamma_j|$ for $i < j$. Some applications of these two special cases are given.

It is shown that various statistics based on the number of runs up and down have an asymptotic multivariate normal distribution under a number of different alternatives to randomness. The concept of likelihood ratio statistics is extended to give a method for deciding what function of these runs should be used, and it is shown that the asymptotic power of these tests depends only on the covariance matrix, calculated under the hypothesis of randomness, and the expected values, calculated under the alternative hypothesis. A general method is given for calculating these expected values when the observations are independent, and these calculations are carried through for a constant shift in location from one observation to the next and for normal and rectangular populations.

Two conservative sample size approximations are given for the Bechhofer formulation of the problem of selecting the population with the largest mean, when the populations have a common known variance. A table of numerical comparisons of these approximations with the exact sample size is included. In addition, both of these results are applied to the problem of selecting the factor with the largest mean from a single multivariate normal population with known covariance matrix.

In this paper it is shown how the systems of partial differential equations developed by the author [15] for the hypergeometric functions of matrix argument may be used to obtain asymptotic expansions for some distributions occurring in multivariate analysis. In particular expansions are derived for the distributions of Hotelling's generalized $T_0^2$-statistic, Pillai's $V^{(s)}$ criterion, and the largest latent root of the sample covariance matrix.

In a special case of Polya's urn scheme, the probability that the first $n$ draws are all of the same color is interpreted as a function of the (single) correlation coefficient. A more general urn model is introduced in which the correlation between pairs of results may differ from pair to pair, and again the probability of consecutive colors is considered. This result is compared with the probability of coincidence in sign under the multivariate normal distribution. The comparison suggests a new approximation for the probability in the multivariate normal case. This approximation appears to be useful only in the Polya case, where the correlations are all equal.

The distribution of the latent roots of the covariance matrix calculated from a sample from a normal multivariate population, was found by Fisher [3], Hsu [6] and Roy [10] for the special, but important case when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I$. By use of the representation theory of the linear group, we are able to obtain the general distribution for arbitrary $\Sigma$.

The paper is divided into two parts: A. An elementary derivation of Bartlett's results on the distribution of the canonical correlation coefficients using exterior differential forms. Briefly, our method consists of taking the original multivariate normal distribution, transforming to the canonical correlations and other variables, and then integrating out these extraneous variables. B. A new method of calculating the conditional moments which appear in Bartlett's expansion of this distribution, based on the process of averaging over the orthogonal group. This method allows the calculation of moments of any order.

The Bayes approach to Multivariate Analysis taken previously by Geisser and Cornfield (JRSS Series B, 1963 No. 2, pp. 368-376) is extended and given a more comprehensive treatment. Posterior joint and marginal densities are derived for vector means, linear combinations of means; simple and partial variances; simple, partial and multiple correlation coefficients. Also discussed are the posterior distributions of the canonical correlations and of the principal components. For the general multivariate linear hypothesis, it is demonstrated that the joint Bayesian posterior region for the elements of the regression matrix is equivalent to the usual confidence region for these parameters. The joint predictive density of a set of future observations generated by the linear hypothesis is obtained thus enabling one to specify the probability that a set of future observations will be contained in a particular region. (Author)

The problem considered is that of finding the "best" linear function for discriminating between two multivariate normal populations, $\pi_1$ and $\pi_2$, without limitation to the case of equal covariance matrices. The "best" linear function is found by maximizing the divergence, $J'(1, 2)$, between the distributions of the linear function. Comparison with the divergence, $J(1, 2)$, between $\pi_1$ and $\pi_2$ offers a measure of the discriminating efficiency of the linear function, since $J(1, 2) \geq J'(1, 2)$. The divergence, a special case of which is Mahalanobis's Generalized Distance, is defined in terms of a measure of information which is essentially that of Shannon and Wiener. Appropriate assumptions about $\pi_1$ and $\pi_2$ lead to discriminant analysis (Sections 4, 7), principal components (Section 5), and canonical correlations (Section 6).

Estimation-preceded-by-testing is studied in the context of estimating the mean vector of a multivariate normal distribution with quadratic loss. It is shown that although there are parameter values for which the risk of a preliminary-test estimator is less than that of the usual estimator, there are also values for which its risk exceeds that of the usual estimator, and that it is dominated by the positive-part version of the Stein-James estimator. The results apply to preliminary-test estimators corresponding to any linear hypothesis concerning the mean vector, e.g., an hypothesis in a regression model. The case in which the covariance matrix of the multi-normal distribution is known up to a multiplicative constant and the case in which it is completely unknown are treated.

In this paper test criteria are developed for testing hypotheses of "compound symmetry" in a normal multivariate population of $t$ variates $(t \geq 3)$ on basis of samples. A feature common to the twelve hypotheses considered is that the set of $t$ variates is partitioned into mutually exclusive subsets of variates. In regard to the partitioning, the twelve hypotheses can be divided into two contrasting but very similar types, and the six in one type can be paired off in a natural way with the six in the other type. Three of the hypotheses within a given type are associated with the case of a single sample and moreover are simple modifications of one another; the remaining three are direct extensions of the first three, respectively, to the case of $k$ samples $(k \geq 2)$. The gist of any of the hypotheses is indicated in the following statement of one, denoted by $H_1(mvc)$: within each subset of variates the means are equal, the variances are equal and the covariances are equal and between any two distinct subsets the covariances are equal. The twelve sample criteria for testing the hypotheses are developed by the Neyman-Pearson likelihood-ratio method. The following results are obtained for each criterion (assuming that the respective null hypotheses are true) for any admissible partition of the $t$ variates into subsets and for any sample size, $N$, for which the criterion's distribution exists: (i) the exact moments; (ii) an identification of the exact distribution as the distribution of a product of independent beta variates; (iii) the approximate distribution for large $N$. Exact distributions of the single-sample criteria are given explicitly for special values of $t$ and special partitionings. Certain psychometric and medical research problems in which hypotheses of compound symmetry are relevant are discussed in section 1. Sections 2-6 give statements of the hypotheses and an illustration, for $H_1(mvc)$, of the technique of obtaining the moments and identifying the distributions. Results for the other criteria are given in sections 7-8. Illustrative examples showing applications of the results are given in section 9.

Many well-known procedures in multivariate data analysis are invariant under the group, $L(p)$, of translations and nonsingular linear transformations. New maximal $L(p)$ invariant statistics are derived and are shown to have the geometrical interpretation of a scatter of points in Euclidean space. The distribution of maximal $L(p)$ invariants for the case of a single multivariate normal population is shown to follow from a result of James (1954). Finally we consider tests of the null hypothesis that $k > 1$ populations are identical and show that optimal $L(p)$ invariant tests are similar tests of randomness.

The response variable of a general multivariate model can be constructed as a positive affine transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the multivariate model has parameters that can be expressed as the mean vector and the variance matrix. In the case of an error variables that is standard normal, it becomes the ordinary multivariate normal. For the general multivariate model with rotationally-symmetric error the sample inner-product matrix is a conventional statistic for inference. The distribution of this statistics is derived: the distribution is a conditional distribution given observable characteristics of the error variable. The response variable of a more specialized multivariable model can be constructed as a positive linear transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the model has parameters that can be expressed in terms of the variance matrix, the mean vector being zero. In the case of an error variable that is standard normal; it becomes the ordinary central multivariate normal. For the multivariate model with rotationally-symmetric error, the distribution of the Wishart matrix is derived; the distribution is a conditional distribution given observable characteristics of the error variable. And for the multivariate model with standard normal error, the noncentral distribution of the Wishart matrix is also derived, again as the appropriate conditional distribution.

As an approximation to the regression function $m$ of $Y$ on $X$ based upon empirical data, E. A. Nadaraya and G. S. Watson have studied estimates of $m$ of the form $m_n(x) = \sum Y_ik((x - X_i)/a_n)/\sum k((x - X_i)/a_n)$. For distinct points $x_1, \cdots, x_k$, we establish conditions under which $(na_n)^{\frac{1}{2}}(m_n(x_1) - m(x_1), \cdots, m_n(x_k) - m(x_k))$ is asymptotically multivariate normal.

Distribution problems in multivariate analysis are often related to the joint distribution of the characteristic roots of a matrix derived from sample observations. This well-known Fisher-Girshick-Hsu-Mood-Roy distribution (under certain null hypotheses) of $s$ non-null characteristic roots can be expressed in the form \begin{equation*}\tag{1.1}f(\theta_1, \cdots, \theta_s) = C(s, m, n) \prod^s_{i = 1} \theta^m_i (1 - \theta_i)^n \prod_{i > j} (\theta_i - \theta_j), 0 < \theta_i \leqq \cdots \leqq \theta_s < 1,\end{equation*} where \begin{equation*}\tag{1.2}C(s, m, n) = \pi^{\frac{1}{2}s^2}\Gamma_s(m + n + s + 1)/ \{\Gamma_s(\frac{1}{2}(2m + s + 1))\Gamma_s(\frac{1}{2}(2n + s + 1))\Gamma_s(\frac{1}{2}s)\}, \end{equation*} $\Gamma_s(\cdot)$ is the multivariate gamma function defined in [2], and $m$ and $n$ are defined differently for various situations described in [4], [6]. Pillai [3], [5] has given the density function of the larger of two roots, i.e. when $s = 2$, as a hypergeometric function. In this paper, the result is extended to the general case giving the density function of the largest of $s$ roots as a generalized hypergeometric function [1], [2]. The density function of the largest root of a sample covariance matrix derived by Sugiyama [7] can be obtained from the one derived here by considering the transformation $\frac{1}{2}\lambda_s = n\theta_s$, and making $n$ tend to infinity.