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.

This paper is concerned with a simple mathematical model for a branching stochastic process. Using the language of family trees we may illustrate the process as follows. The probability that a man has exactly $r$ sons is $p_r, r = 0, 1, 2, \cdots$. Each of his sons (who together make up the first generation) has the same probabilities of having a given number of sons of his own; the second generation have again the same probabilities, and so on. Let $z_n$ be the number of individuals in the $n$th generation. We study the probability distribution of $z_n$. Some previous results are given in section 2; these include procedures for computing moments of $z_n$, and a criterion for when the family has probability 1 of dying out. In sections 3 and 4 the case is considered where the family has a non-zero chance of surviving indefinitely. In this case the random variables $z_n/Ez_n$ converge in probability to a random variable $w$ with cumulative distribution $G(u)$. It is shown that $G(u)$ is absolutely continuous for $u \neq 0$. Results of a Tauberian character are given for the behavior of $G(u)$ as $u \rightarrow 0$ and $u \rightarrow \infty$. In section 5 some examples are given where $G(u)$ can be found explicitly; $G(u)$ is computed numerically for the case $p_1 = 0.4, p_2 = 0.6$. In section 6 families with probability 1 of extinction are considered. A method is given for obtaining in certain cases an expansion for the moment-generating function of the number of generations before extinction occurs. In section 7 maximum likelihood estimates are obtained for the $p_r$ and for the expectation $Ez_1$; consistency in a certain sense is proved. In section 8 a brief discussion is given of the relation between two types of mathematical models for branching processes.

For testing a composite hypothesis, critical regions are determined which are most powerful against a particular alternative at a given level of significance. Here a region is said to have level of significance $\epsilon$ if the probability of the region under the hypothesis tested is bounded above by $\epsilon$. These problems have been considered by Neyman, Pearson and others, subject to the condition that the critical region be similar. In testing the hypothesis specifying the value of the variance of a normal distribution with unknown mean against an alternative with larger variance, and in some other problems, the best similar region is also most powerful in the sense of this paper. However, in the analogous problem when the variance under the alternative hypothesis is less than that under the hypothesis tested, in the case of Student's hypothesis when the level of significance is less than $\frac{1}{2}$, and in some other cases, the best similar region is not most powerful in the sense of this paper. There exist most powerful tests which are quite good against certain alternatives in some cases where no proper similar region exists. These results indicate that in some practical cases the standard test is not best if the class of alternatives is sufficiently restricted.

Let $X$ be the matrix $\lbrack x_{mn}\rbrack, t$ a scalar, and let $\partial X/\partial t, \partial t/\partial X$ denote the matrices $\lbrack\partial x_{mn}/\partial t\rbrack, \lbrack\partial t/\partial x_{mn}\rbrack$ respectively. Let $Y = \lbrack y_{pq}\rbrack$ be any matrix product involving $X, X'$ and independent matrices, for example $Y = AXBX'C$. Consider the matrix derivatives $\partial Y/\partial x_{mn}, \partial y_{pq}/\partial X$. Our purpose is to devise a systematic method for calculating these derivatives. Thus if $Y = AX$, we find that $\partial Y/\partial x_{mn} = AJ_{mn}, \partial y_{pq}/\partial X = A'K_{pq}$, where $J_{mn}$ is a matrix of the same dimensions as $X$, with all elements zero except for a unit in the $m$-th row and $n$-th column, and $K_{pq}$ is similarly defined with respect to $Y$. We consider also the derivatives of sums, differences, powers, the inverse matrix and the function of a function, thus setting up a matrix analogue of elementary differential calculus. This is designed for application to statistics, and gives a concise and suggestive method for treating such topics as multiple regression and canonical correlation.

The paper shows that under very broad conditions the usual theorems concerning the limiting distributions of estimates hold for estimates based on samples selected from finite universes, at random without replacement. It may be remarked that under the same conditions, the same conclusions are true for random sampling from finite universes with replacement, if the universes are permitted to change within the limitations set by condition $W$.

A test is proposed for the independence of two random variables with continuous distribution function (d.f.). The test is consistent with respect to the class $\Omega''$ of d.f.'s with continuous joint and marginal probability densities (p.d.). The test statistic $D$ depends only on the rank order of the observations. The mean and variance of $D$ are given and $\sqrt n(D - ED)$ is shown to have a normal limiting distribution for any parent distribution. In the case of independence this limiting distribution is degenerate, and $nD$ has a non-normal limiting distribution whose characteristic function and cumulants are given. The exact distribution of $D$ in the case of independence for samples of size $n = 5, 6, 7$ is tabulated. In the Appendix it is shown that there do not exist tests of independence based on ranks which are unbiased on any significance level with respect to the class $\Omega''$. It is also shown that if the parent distribution belongs to $\Omega''$ and for some $n \geq 5$ the probabilities of the $n!$ rank permutations are equal, the random variables are independent.

In time series analysis there are two lines of approach, here called the functional and the stochastic. In the former case, the given time series is interpreted as a mathematical function, in the latter case as a random specimen out of a universe of mathematical functions. The close relation between the two approaches is in section 2 shown to amount to a genuine isomorphism. Considering the problem of prediction from this viewpoint, the author gives in sections 3-4 the functional equivalence of his earlier theorem on the decomposition of a stationary stochastic process with a discrete time parameter (see [9], theorem 7). In section 5 the decomposition theorem is applied to the problem of linear prediction. Finally in section 6 a few comments are made. Since various aspects of the isomorphism in question are known, this paper might be regarded as essentially expository.

The Tchebycheff statistical inequality and its generalizations are further generalized so as to apply equally well to $n$-dimensional probability distributions. Comparisons may be made with other generalizations [1], [2] that have been developed recently for the two-dimensional case. The inequalities given in this paper are generally as close as the most favorable corresponding inequalities that exist for the one-dimensional case and in many simple cases they are closer than those that have been given heretofore for two dimensions. In a special case the upper bound of our inequality is actually attained. The theory contains also a less important generalization in one dimension.

In this note are presented graphs of minimum probable population coverage by sample blocks determined by the order statistics of a sample from a population with a continuous but unknown cumulative distribution function (c.d.f.). The graphs are constructed for the three tolerance levels .90, .95, and .99. The number, $m$, of blocks excluded from the tolerance region runs as follows: $m$ = 1(1)6(2)10(5)30(10)60(20)100, and the sample size, $n$, runs from $m$ to 500. Thus the curves show the solution, $\beta$, of the equation $1 - \alpha = I_\beta(n - m + 1, m)$ for $\alpha = .90, .95, .99$ over the range of $n$ and $m$ given above, where $I_x(p, q)$ is Pearson's notation for the incomplete beta function. Examples are cited below for the one- and two-variate cases. Finally, the exact and approximate formulae used in computations for these graphs are given.