The authors study the problem of testing whether the distribution function (d.f.) of the observed independent chance variables $x_1, \cdots, x_n$ is a member of a given class. A classical problem is concerned with the case where this class is the class of all normal d.f.'s. For any two d.f.'s $F(y)$ and $G(y)$, let $\delta(F, G) = \sup_y | F(y) - G(y) |$. Let $N(y \mid \mu, \sigma^2)$ be the normal d.f. with mean $\mu$ and variance $\sigma^2$. Let $G^\ast_n(y)$ be the empiric d.f. of $x_1, \cdots, x_n$. The authors consider, inter alia, tests of normality based on $\nu_n = \delta(G^\ast_n(y), N(y \mid \bar{x}, s^2))$ and on $w_n = \int (G^\ast_n(y) - N(y \mid \bar{x}, s^2))^2 d_yN (y \mid \bar{x}, s^2)$. It is shown that the asymptotic power of these tests is considerably greater than that of the optimum $\chi^2$ test. The covariance function of a certain Gaussian process $Z(t), 0 \leqq t \leqq 1$, is found. It is shown that the sample functions of $Z(t)$ are continuous with probability one, and that $\underset{n\rightarrow\infty}\lim P\{nw_n < a\} = P\{W < a\}, \text{where} W = \int^1_0 \lbrack Z(t)\rbrack^2 dt$. Tables of the distribution of $W$ and of the limiting distribution of $\sqrt{n}\nu_n$ are given. The role of various metrics is discussed.

Incomplete block designs with a few replications are of practical interest to experimenters. Partially balanced incomplete block (PBIB) designs with two associate classes, and $k > r = 2$, were studied by one of the authors [1]. The present paper extends this investigation to the case $k > r$ with $\lambda_1 = 1$ and $\lambda_2 = 0$. It is shown that the parameters of all PBIB designs in this case are given by (4.27) and thus depend upon three integral parameters $k, r,$ and $t$, with the additional restrictions that \begin{equation*}\tag{i}1 \leqq t \leqq r,\end{equation*} \begin{equation*}\tag{i}rk(r - 1)(k - 1)/t(k + r - t - 1)\text{is a positive integer}.\end{equation*} For the particular case $r = 3$ it is shown that all designs with $t = 2$ or 3 necessarily exist, but if $t = 1$, then the only possible value of $k > r$ is 5. However designs with parameters (4.27) with $r = 3, t = 1$, and $k = 2$ or 3 are also combinatorially possible though they do not belong to the class $k > r$. Interesting by-products of this study are a lemma and five corollaries which give an insight into the structure of PBIB designs with $\lambda_1 = 1$ and $\lambda_2 = 0$, and $p^1_{11} = k - 2$, no special assumptions being made regarding $r$ and $k$.

Statistical problems involving angular observations may arise in diverse scientific fields, either from direct measurement of angles--say the direction of winds or of glacial pebbles or of fracture planes--or they may arise from the measurement of times reduced modulo some period and converted into angles--say time of day when train wrecks occur. Specifically, we consider a set of $n$ points $\xi_\nu$ situated on a unit circle and assumed to constitute a sample from a distribution having the p.d.f. $g(\xi)$, where $0 \leqq \xi < 2\pi$. Let the $n$ random unit vectors thus defined have the components $\sin \xi_\nu$ and $\cos \xi_\nu$, and set \begin{equation*}\tag{0.1} V = \sum \cos \xi_\nu,\quad W = \sum \sin \xi_\nu,\quad R = \sqrt{V^2 + W^2}.\end{equation*} Let $P(r, n)$ be the probability that $R \leqq r$, and let $Q = 1 - P(r, n)$. This paper shows how the statistics $V$ and $R$ provide tests for the uniform distribution $g(\xi) = 1/2\pi$. The distribution of $R$ on the hypothesis of uniformity was derived by Kluyver as a solution to Pearson's random walk problem, and is tabulated here for use in significance tests. The distribution of $V$ is derived here, but has not been calculated. To illustrate the type of tests that might employ the statistics $R$ and $V$, consider a carnival wheel--first from the standpoint of the punter who suspects bias, second from the standpoint of the mechanic who has attempted to introduce bias. The punter, by studying the performance of the wheel, might wish to answer two questions: first, does the wheel differ credibly from an unbiased wheel? second, what is the direction and extent of the bias, if any? An answer to the first question is obtainable from the distribution of $R$, and an answer to the second has been provided by Mises. The mechanic, on the other hand--because he knows the direction of the bias, if there is a bias--might better use the statistic $V$ as a test of his success, and he might appropriately modify the Mises approach in estimating the extent of the bias.

Let $x_1, \cdots, x_n$ be independent chance variables with the common distribution function $F(x)$ and the empiric distribution function $F^\ast(x)$. Let $a_n$ be the value of $a$ which minimizes (1) below. In this paper the asymptotic distribution of $\sqrt{n} a_n$ is obtained, subject to certain restrictions on $F(x)$.

The problem is considered of determining the least upper (or greatest lower) bound for the expected value $EK(X_1, \cdots, X_n)$ of a given function $K$ of $n$ random variables $X_1, \cdots, X_n$ under the assumption that $X_1, \cdots, X_n$ are independent and each $X_j$ has given range and satisfies $k$ conditions of the form $Eg^{(j)}_i (X_j) = c_{ij}$ for $i = 1, \cdots, k$. It is shown that under general conditions we need consider only discrete random variables $X_j$ which take on at most $k + 1$ values.

A statistical problem arising in many fields of activity requires the estimation of the average number of events occurring per unit of a continuous variable, such as area or time. The underlying distribution of events is assumed to be Poisson; the constant to be estimated is the unknown parameter $\lambda$ of the distribution. A sampling procedure is proposed in which the continuous variable is observed until a fixed number $M$ of events occurs. Such a procedure enables us to form an estimate $l$, which with confidence coefficient $\alpha$ does not differ from $\lambda$ by more than 100 $\gamma$ per cent of $\lambda$. The values of $\gamma$ and $\alpha$ depend on $M$ but not on $\lambda$. Modifications of this procedure which are sequential in nature and have possible operational advantages are also described. These procedures are discussed in terms of a chemical problem of particle counting. It is clear, however, that they are generally applicable whenever the basic probability assumptions apply.

The main conclusion of this paper can be described as follows. Consider a statistical decision problem in which certain structural conditions are satisfied, and let $T$ be a statistic on the sample space. Then the class of decision functions which depend on the sample point only through $T$ is essentially complete if and only if $T$ is a sufficient statistic. The structural conditions in question are satisfied in many estimation problems.

The initial distribution considered here is obtained from a multivariate analogue of the Pearson Type III distribution, and the value of the correlation is taken to be non-negative. There is obtained here the distribution of the maximum in samples of fixed size $n$ from a random variable which is the arithmetic mean of $k$ such correlated random variables. This distribution is obtained for large values of $n$ and for large values of $k$. The appropriate expressions for the mode and scale parameters are also given.

Tables are presented from which exact values of the Wilcoxon distribution may be obtained when the smaller sample size $m$ does not exceed 12. The Edgeworth approximation to terms of order $1/m^2$ is given and its accuracy investigated.

Our discussion will concern primarily $\lambda$, the ratio of two variances which arise in discussing the "mixed" incomplete block model. In Section 1 we find first the class of invariant statistics for a test involving this ratio and second the joint distribution of these statistics. In Section 2 we use these statistics to construct a test (with certain optimum properties) of the hypothesis $\lambda < \lambda_0$ versus $\lambda > \lambda_1$.

This note expresses the first two moments of the order statistics in samples from the Weibull distribution (sometimes referred to as the "third" asymptotic distribution of extreme values) in terms of known (incomplete $B$ and $\Gamma$) functions. A similar procedure is applied to the "second" asymptotic distribution of extreme values.

Let $(R_1, \cdots, R_n)$ be a random vector which takes on each of the $N!$ permutations of $(1, 2, \cdots, N)$ with equal probability, $1/N!$. Let $(a_{N1}, \cdots, a_{NN})$ and $(b_{N1}, \cdots, b_{NN})$ be two sets of real numbers given for every $N$. We will assume throughout that for no $N$ are the $a_{Ni}$ all equal or the $b_{Ni}$ all equal. We also assume that the $a_{Ni}$ and $b_{Ni}$ have been so normalized that $\sum a_{Ni} = \sum b_{Ni} = 0;\quad \sum a^2_{Ni} = N^{-1} \sum b^2_{Ni} = 1$. Unless otherwise stated, $\sum$ will mean $\sum^N_{i=1}$. Define \begin{equation*}\tag{1.1} S_N = \sum a_{Ni}b_{NR_i}.\end{equation*} Let $\Phi(x)$ be the unit normal c.d.f. In Section 2 sufficient conditions are given for $\mathrm{Pr}\{S_N < x\}$ to approach $\Phi(x)$ as $N \rightarrow \infty$. (The first two moments of $S_N$ are 0 and $N/(N - 1)$, respectively.) For every $N$, let $Y = (Y_{11}, \cdots, Y_{1N_1}, \cdots, Y_{m1}, \cdots, Y_{mN_m})$ be $N = N_1 + \cdots + N_m$ random variables which are mutually independent and independent of the $R_i$. We assume that all $Y_{ij}$ with the same first subscript are identically distributed. Define $\bar{Y} = N^{-1} \sum_{i,j}Y_{ij},\quad S^2 = N^{-1} \sum_{i,j} (Y_{ij} - \bar{Y})^2,\quad SY'_{ij} = Y_{ij} - \bar{Y}, Y'_{ij} = 0 \text{if} S = 0$. Let $Y'$ denote the vector of the $Y'_{ij}$. Let $y = (y_1, \cdots, y_N)$ denote a point in $N$-space. By $F_N(x, y)$ we mean the c.d.f. of the random variable $S_N = \sum a_{Ni}y_{NR_i}$. In Section 3 are considered sufficient conditions for convergence with probability one of the random c.d.f. $F_N(x, Y')$ to $\Phi(x)$.

It is shown that even in very simple situations (like estimating the mean of a normal population) where a uniformly minimum variance unbiased estimator of the unknown population characteristic is known to exist, no best (even locally) unbiased estimator exists as soon as we alter slightly the definition of variance.

The problem of finding a confidence interval of preassigned length and of more than a given confidence coefficient for the unknown mean of a normal distribution with unknown variance is insoluble if the sample size used is fixed before sampling starts. In this paper two-sample plans, with the size of the second sample depending upon the observations in the first sample (as in [1]), are discussed. Consideration is limited to those schemes which increase the center of the final confidence interval by $k$ if each observation is increased by $k$, and for which the size of the second sample is a function only of the differences among the observations in the first sample. Then it is shown that the mean of all the observations taken should be used as the center of the final confidence interval. Those schemes which make the size of the second sample a nondecreasing function of the sample variance of the first sample are shown to have certain desirable properties with respect to the distribution of the number of observations required to come to a decision.