This paper contains an expository discussion of the chi square test of goodness of fit, intended for the student and user of statistical theory rather than for the expert. Part I describes the historical development of the distribution theory on which the test rests. Research bearing on the practical application of the test--in particular on the minimum expected number per class and the construction of classes--is discussed in Part II. Some varied opinions about the extent to which the test actually is useful to the scientist are presented in Part III. Part IV outlines a number of tests that have been proposed as substitutes for the chi square test (the $\omega^2$ test, the smooth test, the likelihood ratio test) and Part V a number of supplementary tests (the run test, tests based on low moments, subdivision of chi square into components).

The most powerful rank order tests against specific parametric alternatives are derived. Following the methods of Hoeffding [4], we derive the most powerful rank order test of whether $N$ observations come from the same but unknown population against the alternative that the observations $Z_1, \cdots, Z_N$ come from populations which have the joint density $\Pi^N_{i = 1} \frac{1}{\sigma\sqrt{2\pi}} \exp \big\lbrack - \frac{1}{2\sigma^2} (z_i - d_i\xi - \eta)^2 \big\rbrack,$ where $d_1, \cdots, d_N$ are given constants, not all equal, and $\xi/\sigma$ is sufficiently small. The test criterion was found to be $c_1(R) = \sum d_iEZ_{N, r_i}$, where $EZ_{Ni}$ is the expected value of the $i$th standard normal order statistic and $R = (r_1, \cdots, r_N)$ is the permutation of the ranks. The distribution of this statistic was shown to be asymptotically normal providing the known constants $d_1, \cdots, d_N$ satisfied Noether's condition [9]. The two-sample distribution is a special case, and the resultant statistic $c_1(R)$ is shown to be asymptotically normal. The approximation of the distribution of the $c_1(R)$ statistic to the distribution $C(1 - x^2)^{\frac{1}{2}N-2}, - 1 \leqq x \leqq 1$, is investigated. This statistic is then compared to the existing Mann and Whitney $U$ statistic. No method having been found for analytical evaluation of the power of this test, the power was examined experimentally. Tables are appended giving the exact distribution of the $c_1(R)$ statistic for all possible subsample sizes whose total size is less than or equal to 10 together with the corresponding Mann and Whitney $U$ value. Table 2 gives critical values of $c_1(R)$ for $N \leqq 10, p \leqq.10$.

Group divisible incomplete block designs are an important subclass of partially balanced designs [1], [2] with two associate classes, and they may also be regarded as a special case of intra- and inter-group balanced incomplete block designs [3], [4]. They may be defined as follows. An incomplete block design with $v$ treatments each replicated $r$ times in $b$ blocks of size $k$ is said to be group divisible (GD) if the treatments can be divided into $m$ groups, each with $n$ treatments, so that the treatments belonging to the same group occur together in $\lambda_1$ blocks and treatments belonging to different groups occur together in $\lambda_2$ blocks. If $\lambda_1 = \lambda_2 = \lambda$ (say), then every pair of treatments occurs in $\lambda$ blocks, and the design becomes a balanced incomplete block design, which has been extensively studied [5], [6], [7], [8]. We shall therefore confine ourselves to the case $\lambda_1 \neq \lambda_2$. The object of this paper is to study the combinatorial properties of these designs. It is shown that the GD designs can be divided into three exhaustive and mutually exclusive classes: (a) Singular GD designs characterized by $r - \lambda_1 = 0$; (b) Semi-regular GD designs characterized by $r - \lambda_1 > 0, rk - v\lambda_2 = 0$; (c) Regular GD designs characterized by $r - \lambda_1 > 0, rk - v\lambda_2 > 0$. Certain inequality relations between the parameters necessary for the existence of the design have been derived in each case. Some other interesting theorems about the structure of these designs have also been obtained. Methods of constructing GD designs will be given in a separate paper.

In this paper we derive the operating characteristic of the control chart for sample means when process standards are unspecified. Under the null hypothesis the distribution of the process is $N(\mu, \sigma^2)$, where $\mu$ and $\sigma$ are fixed but unknown. Under the alternative the process mean is a random variable with a $N(\mu, \theta^2\sigma^2)$ distribution. Exact results are obtained for cases ranging from two samples of size 2 to four samples of 10. Bounds on the operating characteristic are obtained in particular cases ranging from five samples of 5 to 25 samples of 10.

Instead of minimizing the maximum risk it is proposed to restrict attention to decision procedures whose maximum risk does not exceed the minimax risk by more than a given amount. Subject to this restriction one may wish to minimize the average risk with respect to some guessed a priori distribution suggested by previous experience. It is shown how Wald's minimax theory can be modified to yield analogous results concerning such restricted Bayes solutions. A number of examples are discussed, and some extensions of the above criterion are briefly considered.

Sufficient conditions for the existence of a least favorable distribution were given by Wald in his work on general decision theory. It is shown here that for problems of hypothesis testing and more generally for multiple decision problems involving a finite number of decisions, the result holds under a much weaker restriction than Wald's assumption of a compact parameter space.

The concept of "functional sufficiency" or "f-sufficiency" for a class of density functions is introduced and conditions given under which it corresponds to sufficiency as defined by Halmos and Savage [2]. A minimal f-sufficient statistic is defined and shown to exist, and its construction is given. The minimal f-sufficient statistic for a class of densities for which the region of positive density varies with the parameter is shown to be equivalent to the combination of a "statistic of selection" and the minimal f-sufficient statistic for a class of densities for which the region of positive density is fixed. The construction of sufficient statistics in this latter case subject to certain conditions of regularity has been treated by Koopman [1]. If the parameter is a parameter of selection from a fixed distribution, then the statistic of selection is the minimal f-sufficient statistic. If in addition the regions of positive density are monotone and indexed monotonely by a real parameter, then the statistic of selection is sufficient according to the Halmos and Savage definition. Three examples are given to illustrate the results.

In this paper we shall proceed to generalize the notion of a set of orthogonal Latin squares, and we term this extension an orthogonal array of index unity. In Section 2 we secure bounds for the number of constraints which are the counterpart of the familiar theorem which states that the number of mutually orthogonal Latin squares of side $s$ is bounded above by $s - 1$. Curiously, our bound depends upon whether $s$ is odd or even. In Section 3 we give a method of constructing these arrays by considering a class of polynomials with coefficients in the finite Galois field $GF(s)$, where $s$ is a prime or a power of a prime. In the concluding section we give a brief discussion of designs based on these arrays.

The following theorem is proved. If a univariate distribution has moments of first and second order and admits a homogeneous and symmetric quadratic statistic $Q$ which is independently distributed of the mean of a sample of $n$ drawn from this distribution, then it is either the normal distribution ($Q$ is then proportional to the variance) or the degenerate distribution (in this case no restriction is imposed on $Q$) or a step function with two symmetrically located steps (in this case $Q$ is the sum of the squared observations). The converse of this statement is also true.

Let $x_1, x_2, \cdots, x_n$ be positive, identically distributed, independent random variables. It is of some statistical interest to study the distribution of $z_n = (x_1 + x_2 + \cdots + x_n)/\max (x_1, x_2, \cdots, x_n)$. In this paper we give its characteristic function and in a few cases its distribution. A limiting distribution of fairly wide applicability is given in the last section.

In their interesting and pioneering paper Robbins and Monro [1] give a method for "solving stochastically" the equation in $x: M(x) = \alpha$, where $M(x)$ is the (unknown) expected value at level $x$ of the response to a certain experiment. They raise the question whether their results, which are contained in their Theorems 1 and 2, are valid under a condition (their condition (4'), our condition (1) below) which is statistically plausible and is weaker than the condition which they require to prove their results. In the present paper this question is answered in the affirmative. They also ask whether their conditions (33), (34), and (35) (our conditions (25), (26) and (27) below) can be replaced by their condition (5") (our condition (28) below). A counterexample shows that this is impossible. However, it is possible to weaken conditions (25), (26) and (27) by replacing them by condition (3) (abc) below. Thus our results generalize those of [1]. The statistical significance of these results is described in [1].

Let $M(x)$ be a regression function which has a maximum at the unknown point $\theta. M(x)$ is itself unknown to the statistician who, however, can take observations at any level $x$. This paper gives a scheme whereby, starting from an arbitrary point $x_1$, one obtains successively $x_2, x_3, \cdots$ such that $x_n$ converges to $\theta$ in probability as $n \rightarrow \infty$.

Frequency functions of the form (1) below including Pearson's system as a special case are derived from a generating differential equation (2). Their properties are discussed, methods of fitting them are suggested, and their characteristic function is determined.