In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.

Orthogonal arrays can be regarded as natural generalizations of orthogonal Latin squares, and are useful in various problems of experimental design. In this paper the known upper bounds for the maximum possible number of constraints for arrays of strength 2 and 3 have been improved, and certain methods for constructing these arrays have been given.

Suppose that $C$ independent random samples of sizes $n_1, \cdots, n_c$ are to be drawn from $C$ univariate populations with unknown cumulative distribution functions $F_1, \cdots, F_c$. This paper discusses a test of the null hypothesis $F_1 = F_2 = \cdots = F_c$ against alternatives of the form $F_i(x) = F(x - \theta_i)\quad (\text{all} x, i = 1, \cdots, C)$ with the $\theta_i$'s not all equal, or against alternatives of a much more general sort to be specified in Section 5. The test to be discussed has as its critical region large values of the ordinary $F$-ratio for one-way analysis of variance, computed after the observations have been replaced by their ranks in the $\sum n_i$-fold over-all sample. This use of ranks simplifies the distribution theory, and permits application of the test to cases where the ranks are available but the numerical values of the observations are difficult to obtain. Briefly, then, we shall consider a non-parametric analogue, based on ranks, of one-way analysis of variance. It is shown in Section 4 that, under quite general conditions, the proposed test statistic, $H$, is asymptotically chi-square with $C - 1$ degrees of freedom when the null hypothesis holds. Section 5 derives a necessary and sufficient condition that the natural family of sequences of tests based on large values of $H$ all be consistent against a given alternative. Section 6 derives the variance of $H$ under the null hypothesis, Section 7 derives the maximum value of $H$, and Section 8 gives a difference equation which may be used to obtain exact small-sample distributions under the null hypothesis. These derivations are made on the assumption of continuity for the cumulative distribution functions; Section 9 considers extensions to the possibly discontinuous case.

Let the distribution of some random variables depend on real parameters $\theta_1, \cdots, \theta_s$ and consider the hypothesis $H: \theta_i \leqq \theta^\ast_i, i = 1, \cdots, s.$ It is shown under certain regularity assumptions that unbiased tests of $H$ do not exist. Tests of minimum bias and other types of minimax tests are derived under suitable monotonicity conditions. Certain related multidecision problems are discussed and two-sided hypotheses are considered very briefly.

A class of decision problems concerning $k$ populations was considered in [1] and it was shown that a particular decision rule is the uniformly best `impartial' decision rule for many problems of this class. The present paper provides certain improvements of this result. The authors define impartiality in terms of permutations of the $k$ samples rather than in terms of the $k$ ordered values of an arbitrarily chosen real-valued statistic as in the earlier paper. They point out that (under conditions which are satisfied in the standard cases of $k$ independent samples of equal size) if the same function is a sufficient statistic for each of the $k$ samples then the conditional expectation of an impartial decision rule given the $k$ sufficient statistics is also an impartial decision rule. A characterization of impartial decision rules is given which relates the present definition of impartiality with the one adopted in [1]. These results, together with Theorem 1 of [1], yield the desired improvements. The argument indicated here is illustrated by application to a special case.

It sometimes happens that of two empirical cumulative distribution curves (step curves) one lies entirely above the other, in other words that, except at both ends, they have no point in common. The problem then arises, what is the probability that this will happen when both are random samples from the same population. In this paper a partial answer will be given, based on the ingenious solution of Andre (as cited in the well known textbook of Bertrand [1] in the problem of the ballot and also in Chap. VIII, Sect. 5 of [7]). Moreover an analogous method will allow us to give an exact answer to the problem of the maximum difference between two empirical cumulative distribution functions of random samples from the same population, but only if both samples have the same size. Smirnov has given an asymptotic solution for the latter problem (cited by Feller [3], see also [2]). Our result leads, by using the Stirling approximation for the factorials, to the asymptotic formula of Smirnov. A comparison of numerical results of the exact formula and the asymptotic formula of Smirnov shows that at least in the case of equal samples, the probabilities calculated by the Smirnov formula have, for samples as small as 20, an error of less than 4% for probabilities 0.033 or more. (See also Massey [5], who has calculated the exact probabilities for equal samples by means of difference equations.)

Professor John Tukey suggested the following two problems to the author: given that $X_1, X_2, \cdots, X_n$ are normally and independently distributed with unknown means $\mu_1, \mu_2, \cdots, \mu_n$ and given variance $\sigma^2$; PROBLEM A: Find a $\beta$-level confidence interval of the form $g(x_1, \cdots, x_n) \geqq \mu_1, \cdots, \mu_n \geqq - \infty.$ PROBLEM B: Find a $\beta$-level confidence interval of the form $g(x_1, \cdots, x_n) \geqq \mu_1, \cdots, \mu_n \geqq h(x_1, \cdots, x_n).$ The main result of the paper is the nonexistence of intervals satisfying mild regularity conditions and having an exact confidence level (unless $n = 1$ or $\beta = 0, 1)$. However for each problem an interval is given for which the confidence level is greater than or equal to $\beta$ (formulas (2.1), (4.1)); these intervals are apparently shorter than those previously used in practice. Also the procedure for obtaining any interval with at least $\beta$ confidence is described. Some results are discussed for distributions other than the normal.

This paper is concerned with sequential minimax estimation of the parameter $\theta(0 < \theta < \infty)$ of the density function (3.1) when the observations are independently and identically distributed with this density, each observation costs the same amount $c > 0$, and the weight function is as given in Section 2. A procedure requiring a fixed sample size is shown to be a minimax solution for this problem.

Results are given whereby the methods described in an earlier paper [1], dealing with the parametric case, may be applied also to the case of random, or mixed random and parametric components.

It is well known that if every pair of treatments in a symmetrical balanced incomplete block design occurs in $\lambda$ blocks, then every two blocks of the design have $\lambda$ treatments in common. In this paper it will be shown that a somewhat similar property holds for symmetrical group divisible designs. In the course of the investigation there will be introduced certain matrices which are of intrinsic interest.

A slippage problem for normal distributions is formulated as a multiple decision problem, and a solution is obtained which has certain optimum properties. The discussion is confined to the fixed sample case with the same number of observations from each distribution, and the normal distributions involved are assumed to have a common but unknown variance.

A multidimensional analogue of the von Mises statistic is considered for the case of sampling from a multidimensional uniform distribution. The limiting distribution of the statistic is shown to be that of a weighted sum of independent chi-square random variables with one degree of freedom. The weights are the eigenvalues of a positive definite symmetric function. A modified statistic of the von Mises type useful in setting up a two sample test is shown to have the same limiting distribution under the null hypothesis (both samples come from the same population with a continuous distribution function) as that of the one-dimensional von Mises statistic. We call the statistics mentioned above von Mises statistics because they are modifications of the $\omega^2$ criterion considered by von Mises [5]. The paper makes use of elements of the theory of stochastic processes.

In [1] a method for removing the regularity conditions from the Cramer-Rao Inequality was given and applied to the estimation of a single real parameter. It was noted there that the method would extend to problems more general than estimating a single real parameter. However, the method extends also for the estimation of a single real parameter and produces analogues of the Bhattacharyya bounds with and without nuisance parameters.

Suppose we have $k$ lists of names, no name appearing more than once in each list. We are interested in estimating the following parameters: (a) the number of names occurring in common in pairs, triples, $\cdots$, of lists; (b) the number of names occurring in $1, 2, \cdots, k$ lists. This note presents unbiased estimators for these parameters when a random sample is drawn from each list. It is also observed that the estimators presented are the only real-valued statistics which are unbiased estimators of the parameters, and hence must be the minimum variance unbiased estimators. This yields another example in which "insufficient" statistics have been used to obtain minimum variance unbiased estimators. These unbiased estimators may at times give unreasonable estimates. In such cases, it is suggested that the statistics be modified so that the nearest reasonable estimate is used. Although this procedure introduces some bias, it usually reduces the mean square error. This problem arises when we are interested in tracing the interrelations of agencies through the individual members. The problem also arises in the work of H. H. Fussler and J. M. Dawson of the University Library, University of Chicago, who are interested in comparing the acquisitions of various libraries. For special problems other sampling schemes may be more economical or more efficient than taking a sample from each list. Professor F. F. Stephan of Princeton University pointed out to the author that, in the special case of the "library problem," the Book Catalog and author cards used by many libraries provide a convenient means of drawing matched samples. (There is a brief discussion of this kind of sampling problem on page 571 of [1].) A sampling scheme based on the last digit or two of the serial number of the cards could be used to search each library reference file for the same list of books. Special provision must be made for accessions made outside the sampling period and for books not covered by the Library of Congress cards. The analysis presented herein deals with the case in which (either for good, bad, or no reasons) a random sample has been drawn from each list. The restriction that no name appear more than once in each list may be weakened to obtain somewhat more general results. The problem discussed in this paper was brought to the author's attention by Professor W. Allen Wallis of the University of Chicago.*