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.

In this paper there are developed for the first time analytical methods for the investigation of the structure of unsymmetrical balanced incomplete block designs. Two unsymmetrical balanced incomplete block designs are proved to be impossible, and for such designs in general, inequalities are found for the number of treatments common to two blocks.

When an infinite lot consisting of defective and nondefective items is investigated by means of a group sequential sampling plan, the use of matrices and vectors is helpful in determining the probabilities of various combinations of the two classes of items and in computing unbiased estimates of the lot fraction defective. For a sequential plan of the Bartky [1] type, the infinite summation of such vectors leads to an exact, explicit formula for the average number of items inspected, \begin{equation*}\tag{1.1}\bar n_p = p^{-1}\{L_p\lbrack G(h_1 + h_2 - 1) - (h_1 + h_2)\rbrack - G(h_2 - 1) + h_1 + h_2 - \lbrack h_1\rbrack\}\end{equation*}, where $p$ is the fraction defective in the lot, $L_p$ is the probability of arriving at a decision to accept the lot, $h_1$ and $h_2$ are parameters of the plan as defined by the Statistical Research Group [3], $G(i)$ is defined by Bartky's equation (36), and $\lbrack h_1\rbrack$ is the largest integer equal to or less than $h_1$. In approximating $L_p$, or in finding the parameters of a sequential plan with specified risks, the formulas proposed by Wald [2] and the Statistical Research Group can be improved by adding an adjustment, \begin{equation*}\tag{1.2}a = \frac{1}{3}(1 - 2s)\end{equation*}, to the value of $h_2$ wherever it occurs. Their formula for approximating $\bar n_p$ can be improved by adding the adjustment \begin{equation*}\tag{1.3} cq = aq/(1 - s)\end{equation*} wherever $h_2$ occurs, provided that the value of $L_p$ which appears in this formula is arrived at by employing adjustment (1.2).

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).

Correct factors to the probabilities that the two-sample $t$ and $F$ statistics shall exceed fixed positive values $t_0$ and $F_0$ either numerically or arithmetically and to the probabilities that $t$ shall be exceeded by fixed negative values $t_0$ have been derived geometrically. The derivations, when the population has a normal density function, produce a novel method of obtaining the usual distributions of $t$ and of $F$. The correct factors permit the use of existing tables and the results are asymptotically correct for numerically large values of the test statistics. There is some indication that the correction factors are better for small sample sizes than large ones. Useful bounds on the errors committed by using the asymptotic corrections at usual significance levels have not been obtained and are a subject for future investigation. For difficult cases a method of approximate evaluation of the correction factors is provided.

This paper deals with a class of statistical quality control procedures and continuous inspection procedures which are optimum for a specified income function and a production model which can only be in one of four states, two of which are states of repair, with known transition probabilities. The Markov process, generated by the model and the class of decision procedures, approaches a limiting distribution and the integral equations from which the optimum procedures can be derived are given.

The use of the range in place of the standard deviation as a measure of dispersion has long been recognized as a convenient and easily calculated statistic. Possibly its most notable employment has been in the industrial statistician's quality control charts. A statistic based upon the range is described below which may be used to test whether one of a group of means may be considered to be a straggler from all or some of the others in a two-way analysis of variance.