It has been shown in [1] that all partially balanced incomplete block (PBIB) designs with two associate classes, can be divided into a small number of types according to the nature of the association relations among the treatments. One simple and important type is the group divisible (GD). The combinatorial properties of GD designs have been studied in [2] and the analysis along with that for other types is given in [1]. Here we give methods of constructing GD designs. These designs are likely to prove useful in agricultural, genetic and industrial experiments.

This paper describes briefly the current mathematical models upon which communication theory is based, and presents in some detail an exposition and partial critique of C. E. Shannon's treatment of one such model. It then presents a general limit theorem in the theory of discrete stochastic processes, suggested by a result of Shannon's.

In this paper two closely related heuristic principles of test construction (to be explained in Section 3), called Type I and Type II methods, of which Type II is identified with the usual likelihood ratio method, are noticed as underlying most of the classical tests of hypotheses, simple or composite, on means of univariate normal populations, and on total or partial correlations or regressions in the case of multinormal variates. In these situations the two methods are found to lead to identical tests having properties which happen to be very good in certain cases and moderately good in others. For certain types of composite hypotheses an extension is then made of the Type I method which is applied to construct tests of three different classes of hypotheses on multinormal populations (so as to cover, between them, a very large area of multivariate analysis), yielding in each case a test in general different from the corresponding and current likelihood ratio test. In each case, however, the two tests happen to come out identical for some degenerate "degrees of freedom." In contrast to the likelihood ratio test it is found that in these cases, for general "degrees of freedom," the corresponding Type I test is much easier to use on small samples, because of the relatively greater simplicity of the corresponding small sample distribution problem under the null hypothesis. In each case a lower bound of the power function of the Type I test is also given (against all relevant alternatives), anything like which, so far as the author is aware, would be far more difficult to obtain for the Type II tests in these situations. In this paper the general approach to the two methods is entirely of a heuristic nature except that, under fairly wide conditions, a lower bound to the power functions for each of the two types of tests is indicated to be readily available, which, however, is much too crude or wide a bound in general.

Let $X_1, X_2, \cdots, X_n$ be $n$ independent random variables each distributed uniformly over the interval (0, 1), and let $Y_0, Y_1, \cdots, Y_n$ be the respective lengths of the $n + 1$ segments into which the unit interval is divided by the $\{X_i\}$. A fairly wide class of statistical problems is related to finding the distribution of certain functions of the $Y_j$; these problems are reviewed in Section 1. The principal result of this paper is the development of a contour integral for the characteristic function (ch. fn.) of the random variable $W_n = \sum^n_{j=0} h_j(Y_j)$ for quite arbitrary functions $h_j(x)$, this result being essentially an extension of the classical integrals of Dirichlet. The cases of statistical interest correspond to $h_j(x) = h(x),$ independent of $j$. There is a fairly extensive literature devoted to studying the distributions for various functions $h(x)$. By applying our method these distributions and others are readily obtained, in a closed form in some instances, and generally in an asymptotic form by applying a steepest descent method to the contour integral.

The purpose of the present paper is to contribute to the sequential theory of testing hypotheses about stochastic processes with a continuous parameter (say, $t$ which one may think of as time). Sequential decision problems about such processes seem not to have been treated before. Subsequently we shall treat problems of point and interval estimation and general sequential decision problems for such processes. The results, in addition to their interest per se and their practical importance, also shed light on the corresponding results for discrete stochastic processes. The subjects of sequential analysis and the theory of decision functions were founded by Wald, and we treat our present subjects in the spirit of his approach. The general results of decision theory, such as the complete class theorem, carry over to sequential problems about stochastic processes with continuous time parameter. As specific examples we treat the Wiener and Poisson processes and obtain, for example, the exact power function. (For discrete processes the corresponding known results, due to Wald, are approximations).

Sherman [8] and Stein [9] have shown that a method given by the author [1] for comparing two experiments is equivalent, for experiments with a finite number of outcomes, to the original method introduced by Bohnenblust, Shapley, and Sherman [4]. A new proof of this result is given, and the restriction to experiments with a finite number of outcomes is removed. A class of weaker comparisons--comparison in $k$-decision problems--is introduced, in three equivalent forms. For dichotomies, all methods are equivalent, and can be described in terms of errors of the first and second kinds.

For the problem of testing one simple hypothesis against another, of all tests whose probabilities of incorrectly accepting the first hypothesis and of incorrectly accepting the second hypothesis are bounded from above by given bounds, the familiar Wald sequential probability ratio test gives the smallest expectation of sample size under either hypothesis. In this paper, a "generalized sequential probability ratio test" is introduced which differs from the Wald test only in that the same limits ($A, B$ in the usual notation) are not necessarily used at each stage of the sampling, but at the $i$th stage $A_i$ and $B_i$ are used, where these numbers are predetermined constants. It is shown that for any given test $T$, there is a generalized sequential probability ratio test $G$ whose probabilities of incorrectly accepting either hypothesis are no larger than the corresponding probabilities for $T$, and such that the cumulative distribution function of the number of observations required to come to a decision when using $G$ is never below the corresponding distribution function when using $T$, under either hypothesis. We may then say that "$G$ is uniformly better than $T$."

This paper develops explicit closed formulas for the covariances of order statistics in samples from the extreme-value distribution which involve only tabulated functions. Such results do not appear to have been given previously. They have been used in an investigation of the estimation of extreme-value parameters by means of order statistics which will be presented in a fuller report to be submitted to the National Advisory Committee for Aeronautics.

In a recent paper [2] the author developed a discrete distribution and several derived limiting distributions for the number of "hits" on a $k$-dimensional Gaussian target. The purpose of the present paper is to apply these results to the two-dimensional problem considered by Cunningham and Hynd [1]. A general expression and two limiting forms are obtained for the probability of at least one hit. The numerical evaluation using the data in [1] is considered for $n = 5$ rounds, and the probability of at least one hit is plotted in Fig. 1 for various combinations of aiming and dispersion error. For a given over-all time interval the evaluation for large $n$ is discussed in Section 5 and illustrated using the data from [1].

Fixed sample-size procedures are considered for testing a one-sided composite hypothesis concerning a real, one-dimensional parameter of an exponential distribution (1.1). In particular, conditions are studied such that the minimax tests have a critical region which is a semi-infinite interval on the real line.

The present note calls attention to a simple algorithm for computation of the orthogonal matrix associated with the matrix of the normal equations of least squares. An application of the "forward" solution associated with the orthogonalization process is also pointed out.

Let $(R_1, \cdots, R_N)$ be a random vector which takes on each of the $N!$ permutations of the numbers $(1, \cdots, N)$ with equal probability, $1/N!$. Sufficient conditions are given for the asymptotic normality of $S_N = \sum^N_{i=1}a_{Ni}b_{NR_i}$, where $(a_{N1}, \cdots, a_{NN}), (b_{N1}, \cdots, b_{NN})$ are two sets of real numbers given for every $N$. These sufficient conditions are apparently quite different from those given by Wald and Wolfowitz [9] and extended by various writers [4, 7]. In some situations the conditions given here may be easier to apply than those given previously. The most general conditions available to date appear to be those of Hoeffding [4]. In the examples below, however, is given a case of an $S_N$ which does not satisfy the conditions required by Hoeffding's theorem but which is asymptotically normal by our results.

Mills [1], Gordon [2], Birnbaum [3], and the author [4] have studied the ratio of the area of the standardized normal curve from $x$ to $\infty$ and the ordinate at $x$. The object of this note is to establish the monotonic character of, and to obtain lower and upper bounds for, the ratio of the ordinate of the standardized Type III curve at $x$ and the area of the curve from $x$ to $\infty$. This ratio, as shown by Cohen [5] and the author [6], has to be calculated for several values of $x$ when solving approximately the equations involved in the problem of estimating the parameters of Type III populations from truncated samples. It was found by the author that, for large values of $x$, when the ordinates and areas are small, either this ratio cannot be obtained from existing tables prepared by Salvosa [7] or that very few significant digits are available for its calculation. It was thus found desirable to obtain lower and upper bounds which could satisfactorily locate this ratio. The monotonic behavior of this ratio and the inequalities obtained may also prove useful in checking the accuracy of the tables in [7], and in studying the nature of the tail of the chi square distribution.