The general canonical correlation distribution is given as a multiple power series in the true canonical correlations $\rho_i$. When only one true correlation is not zero, this series is expressible as a generalized hypergeometric function, for the cases both of non-central means and of correlations proper. In the general case of more than one non-zero true correlation the coefficients in the expansion depend on the conditional moments of the sample correlations between the pairs of transformed variables representing the true canonical variables, when the sample canonical correlations between the sample canonical variables are fixed. Methods are given of obtaining these coefficients for both cases, non-central means and correlations proper; and their form up to the fourth order, corresponding to $O(\rho^8)$ in the expansion, listed in Appendix I. The detailed terms making up these coefficients are given, in the case of two non-zero correlations, up to the fourth order, and in the general case, up to the third order, in Appendix II.

Although there exists voluminous literature on the theory of probability of independent events, and powerful techniques have been developed for the analysis of most of the interesting problems in this field, the theory of probability of dependent events has been rather neglected. The first detailed investigations in this subject were published by A. Markoff [1]. S. Bernstein [2] has extended the fundamental limit theorems to chains of dependent events. The most extensive exposition of this field has been made by M. Frechet [3]. In the present paper we shall develop methods of averaging functions over chains of dependent variables and find the probability distribution of these functions. It will be shown that for certain types of chains these averages and distribution functions can be expressed in terms of the characteristic values and vectors of a certain operator equation. Many of the methods discussed here have been applied to problems in statistical mechanics [4,5,6,7,8]. The most important application has been made by L. Onsager [8] who proved rigorously (on the basis of a simplified model) that Boltzmann's energy distribution in a solid with cooperative elements leads to a phase transition. The first explicit application of linear operator theory (through matrices and integral equations) to probability chains has apparently been made by Hostinsky [9].

Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.

The purpose of this paper is to give a few theorems concerning the reciprocal relation between the convergence of a sequence of distribution functions and the convergence of the corresponding sequence of their moment generating functions. The paper consists of two parts. In the first part the univariate case is discussed. The content of this part is closely related to that of a recent paper by J. H. Curtiss [1, p. 430-433], but the results are of a somewhat more general nature, and the methods of proofs are different and do not make use of the theory of a complex variable. The second part deals with the multivariate case which, as far as the author knows, has not been treated before with proofs in as complete and rigorous a way. In both the univariate and multivariate cases the proofs are based on the well known Helly selection principle [2, p. 26] for bounded sequences of monotonic functions.

It is desired to find an approximate distribution of simple form for the statistic $\bar r = \frac{x_1x_2 + \cdots + x_Tx_1} {x_1^2 + \cdots + x_T^2}$ ($\bar r$ is an estimate of the serial correlation coefficient $\rho$ in a circular universe) in the case that $\rho \neq O$ in the universe. Such a distribution is obtained by smoothing the joint characteristic function of the numerator and denominator of the expression for $\bar r$. The first two moments are calculated; from these $\bar r$ is seen to be a consistent estimate of $\rho$. A graph of this distribution for sample size $T = 20$ and various values of $\rho$ is given. In addition, an approximate distribution for $p = x^2_1 + \cdots + x^2_T$ is derived which reduces to the exact $(\chi^2-)$ distribution if $\rho = 0$. From a formula which yields all moments, it is concluded that, at least up to the degree of approximation attained, $p/T$ is an unbiased and consistent extimate of $\sigma^2$.

In practical applications it is frequently assumed that the values obtained by a sampling process are independently drawn from the same normal population. Then confidence intervals and significance tests which were derived under the assumption of independence are applied using these values. Often the assumption of independence between the values may be at best only approximately valid. For some cases, however, it may be permissible to assume that the correlation between each two values is the same (intraclass correlation). The purpose of this paper is to investigate the effect of this intraclass correlation on the confidence coefficients and significance levels of several well known confidence intervals and significance tests which were derived under the assumption of independence, and to extend these considerations to the case of two sets of values. In the first part of the paper the relations given in Table I are used to compute tables which show the effect of intraclass correlation on the confidence coefficients and significance levels of the confidence intervals and significance tests listed in Table II. The second part of the paper consists of the proofs of the relations given in Table I.

For each hypothesis $H$ of a certain class of simple hypotheses, a family $F$ of tests is determined such that (a) given any test $w$ of $H$ there exists a test $w'$ belonging to $F$ which has power uniformly greater than or equal to that of $w$. (b) no member of $F$ has power uniformly greater than or equal to that of any other member of $F$. The effect on $F$ of various assumptions about the set of alternatives are considered. As an application an optimum property of the known type $A_1$ tests is proved, and a result is obtained concerning the most strigent tests of the hypotheses considered.

It is shown that $E\lbrack f(x) E(y \mid x)\rbrack = E(fy)$ whenever $E(fy)$ is finite, and that $\sigma^2E(y \mid x) \leq\le \sigma^2y$, where $E(y \mid x)$ denotes the conditional expectation of $y$ with respect to $x$. These results imply that whenever there is a sufficient statistic $u$ and an unbiased estimate $t$, not a function of $u$ only, for a parameter $\theta$, the function $E(t \mid u)$, which is a function of $u$ only, is an unbiased estimate for $\theta$ with a variance smaller than that of $t$. A sequential unbiased estimate for a parameter is obtained, such that when the sequential test terminates after $i$ observations, the estimate is a function of a sufficient statistic for the parameter with respect to these observations. A special case of this estimate is that obtained by Girshick, Mosteller, and Savage [4] for the parameter of a binomial distribution.

Both population and sample mean distributions can be represented or approximated by Pearson curves if the first four moments of the population are finite. Using the $\alpha^2_3, \delta$ chart of Craig [2] to determine the Pearson curve type for the population, an analogous $\bar \alpha^2_3, \delta$ chart is derived for the distribution of the mean. This defines a one to one transformation of $\alpha^2_3, \delta$ into $\bar \alpha^2_3, \bar \delta$. The properties of this transformation are used to discuss the approach to normality of the distribution of the mean as dictated by the central limit theorem. This is facilitated by superposing on the $\alpha^2_3, \delta$ chart the $\bar \alpha^2_3, \bar \delta$ charts for samples of 2, 5, and 10.

In many natural assemblies of elements, the probability of an event for a given element depends not only on the intrinsic nature of that particular element, but also on the states of some or all of the rest of the elements belonging to the same assembly. On the basis of this general idea of "contagion" some urn schemes are developed in this paper in which one has contagious influence in space and time. The most interesting result found is that in general the points of convergence of the probability of the assembly are given by some of the roots of an equation $p = f(p)$ and that some of these roots, between zero and one, represent stable states of the assembly, or points of convergence, and others represent unstable ones, or points of divergence. The two neighboring roots, (if they are single), of a root representing a point of convergence are unstable values of the probability. Consequently, under certain conditions, the limiting probability may be made to have a finite jump by changing the initial probability by an arbitrarily small amount. The concrete cases developed in this paper can be considerably extended by similar methods by assuming more complicated and general assemblies and laws of contagion.