The paper investigates the power of a family of nonparametric tests which includes those known as tests based on permutations of observations. Under general conditions the tests are found to be asymptotically (as the sample size tends to $\infty$) as powerful as certain related standard parametric tests. The results are based on a study of the convergence in probability of certain random distribution functions. A more detailed summary will be found at the end of the Introduction.

The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty<x<\infty} \{| F(x) - F_n(x) | \psi^\frac{1}{2}\lbrack F(x) \rbrack\}$ and $n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x).$ A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including $\psi = 1$ and $\psi = 1/\lbrack t(1 - t) \rbrack$ we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises $\omega^2$ criterion is given.

Given $m + n$ real integrable functions $f_1, \cdots, f_m, g_1, \cdots, g_n$ of a point $x$ in a Euclidean space $X$, a real function $\phi(z_1, \cdots, z_n)$ of $n$ real variables, and $m$ constants $c_1, \cdots, c_m$, the problem considered is the existence of a set $S^0$ in $X$ maximizing $\phi\big(\int_s g_1 dx, \cdots, \int_s g_n dx\big)$ subject to the $m$ side conditions $\int_s f_i dx = c_i$, and the derivation of necessary conditions and of sufficient conditions on $S^0$. In some applications the point with coordinates $\big(\int_s g_1 dx, \cdots, \int_s g_n dx\big)$ may also be required to lie in a given set. The results obtained are illustrated with an example of statistical interest. There is some discussion of the computational problem of finding the maximizing $S^0$.

In this paper we consider the problem of estimation of parameters from a sample in which only the first $r$ (of $n$) ordered observations are known. If $r = \lbrack qn \rbrack, 0 < q < 1$, it is shown under mild regularity conditions, for the case of one parameter, that estimation of $\theta$ by maximum likelihood is best in the sense that $\hat{\theta}$, the maximum likelihood estimate of $\theta$, is (a) consistent, (b) asymptotically normally distributed, (c) of minimum variance for large samples. A general expression for the variance of the asymptotic distribution of $\hat{\theta}$ is obtained and small sample estimation is considered for some special choices of frequency function. Results for two or more parameters and their proofs are indicated and a possible extension of these results to more general truncation is suggested.

This paper investigates certain statistical problems arising in the determination of the ``best'' of $k$ categories when comparing $k - 1$ experimental categories with a standard or control. The discussion is limited to the case of a single stage sampling procedure with an equal number of observations on each of the $k$ categories. Results both of an exact and of an approximate nature are obtained when (a) the observations with each category are normally distributed, and (b) the observations with each category have a binomial distribution.

For quality control charts controlling the mean of a population either small samples may be taken out at frequent intervals or larger samples at less frequent intervals. In this paper, a simple formula is derived by which the most suitable sample size can be determined, leading to the detection of any given change of the population mean with a minimum of inspection.

If for the estimation of $\beta_1, \beta_2$ different observations (``sources'') of form (1.1) are potentially available, each of them being repeatable as many times as we please, the question arises which of them the experimenter should utilize, and in what proportions. With appropriate optimality conventions the answer is the following. For the estimation of a single quantity of form $\theta = \alpha_1\beta_1 + \alpha_2\beta_2$ the optimum allocation comprises two sources only; for the estimation of both parameters, the corresponding number is two or three; the best proportions are indicated in Sections 2 and 4 below. Generalizations to more than two parameters and to observations at different costs are briefly discussed. The problem is related to Hotelling's weighing problem [2] and to the topics treated by David and Neyman in [1].

Two classification statistics discussed in the literature can be written as functions of the elements of a $2 \cdot 2$ symmetric random matrix $M$. An analytic derivation is given of the distribution of $M$, and of a related matrix $M^\ast$, extending earlier work on distribution theory by Wald [1] and Anderson [2].

The exact chance distribution of the number of isolates in a social group is found in this paper, using methods due to Frechet. The binomial distribution fitted to the first two moments of the exact distribution is shown to give reasonably good approximation and a slightly coarser binomial approximation is indicated.

Doob [1] has given heuristically an appealing methodology for deriving asymptotic theorems on the difference between the empirical distribution function calculated from a sample and the actual distribution function of the population being sampled. In particular he has applied these methods to deriving the well known theorems of Kolmogorov [2] and Smirnov [3]. In this paper we give a justification of Doob's approach to these theorems and show that the method can be extended to a wide class of such asymptotic theorems.

The greatest lower bound os the $n$th probability moment (1.1) of a population with variance $\sigma^2$ is given by (3.4).

In 1922 MacNeish [1] considered the problem of orthogonal Latin squares and showed that if the number $s$ is written in standard form: $s = p^{n_0}_0p^{n_1}_1 \cdots p^{n_k}_k,$ where $p_0, p_1, \cdots, p_k$ are primes, and if $r = \min(p^{n_0}_0, p^{n_1}_1, \cdots, p^{n_k}_k),$ then we can construct $r - 1$ orthogonal Latin squares of side $s$. An alternative proof was also given by Mann [2]. At the April, 1950 meeting of the Institute of Mathematical Statistics at Chapel Hill, North Carolina, R. C. Bose announced an interesting generalization of this result [3] which is stated as a theorem in the next section. The proof given here is simpler than Bose's original proof and is published at his suggestion.