The terminology is defined and illustrated in Section 1. A little historical background not very familiar to statisticians is sketched in Section 2. In Section 3 some difficulties about the formulation of random interactions are discussed. Section 4 deals with models reflecting a randomization in the experiment to assign the treatment combinations to finite populations of experimental units.

In many statistical decision problems, the observations can be summarized in a single sufficient statistic such that the likelihood ratio for any two distributions in the family under consideration is a monotone function of that statistic. This paper assumes, accordingly, that the statistician's decision is to be based upon a single observation of a random variable $X$, whose distribution is given by (1) and satisfies the inequality (2) in Section 1. As examples of this family of distributions, we have the exponential family such as the normal, binomial, and Poisson. Other kinds of examples are given in Section 1. In connection with the ordinary testing problem, Allen [1] showed that for the composite testing problem of the one-sided type for the special case of the exponential family of distributions, an admissible minimax procedure must be of the form: choose action 1 (accept the hypothesis) if $x < x_0$ and choose action 2 (accept the alternative) if $x > x_0$. If $x = x_0$, randomization may be required. Sobel [2] and Chernoff obtained partial results for the same class of distributions when the set of decisions is finite. This paper unifies, extends, and strengthens these results and treats of a wide variety of statistical decision problems for which the densities have a monotone likelihood ratio. In Section 1 the fundamental definition and preliminaries are introduced. In particular, the conditions imposed on the loss functions and the densities are delimited and some simple properties of these quantities are developed. In Section 2 we establish some of the basic lemmas. Noteworthy are Lemmas 1 and 2 which express the variation of sign diminishing properties of the densities which possess a monotone likelihood ratio. The essential completeness of the set of all monotone strategies (see Section 3 for the definition) in the class of all statistical procedures is demonstrated in Section 3 for the case of a finite number of actions. Section 4 deals with the problem of determining the form of all Bayes strategies for the statistician. The important problem of admissibility is studied in detail in Section 5. In the next section a study of the Bayes strategies for nature is made for the case of two actions. In Section 7 the complete class theory is carried through for the case of an infinite number of actions. This is accomplished by employing an argument involving a limiting procedure from the case of finite actions as treated in Section 3. The eighth section presents an analysis of the nature of the Bayes strategies for the case of an infinite number of actions. The final section entails a brief discussion of the connection of invariance theory and the conditions of monotonicity as are required throughout this paper. Further extensions of these ideas in a different direction, which involves relaxing the conditions on the loss functions and strengthening the requirements on the densities, can be found in [3].

Let $F_{\lambda^0}$ denote the joint distribution of two independent random variables $Y_{\lambda^0}$ and $Z_{\lambda^0}$. The paper investigates properties of the joint distribution $F_\lambda$ of the linearly transformed random variables $Y_\lambda$ and $Z_\lambda$. Let $\Im_0$ be the Spearman rank correlation test, $\Im_1$ the difference sign correlation test, $\Im_2$ the unbiased grade correlation test (which is asymptotically equivalent to $\Im_0$), $\Im_3$ the medial correlation test, and $\mathcal{R}$ the ordinary (parametric) correlation test. (Whenever discussing $\mathcal{R}$ we assume existence of fourth moments.) Properties of the power of these tests are found for alternatives of the above-mentioned form, particularly for alternatives "close" to the hypothesis of independence and for large samples. Against these alternatives the efficiency of $\Im_3$ is found to depend strongly on local properties of the densities of $Y_{\lambda_0}$ and $Z_{\lambda^0}$, which should invite caution; and the efficiency of $\Im_1$ with respect to $\Im_0$ is often unity. Incidentally, Pitman's result on efficiency is extended in several directions.

Consider samples from continuous distributions $F(x)$ and $F(x - \theta)$. We may test the hypothesis $\theta = 0$ by using the two-sample Wilcoxon test. We show in Section 1 that its asymptotic Pitman efficiency, relative to the $t$-test, never falls below 0.864. This result also holds for the Kruskal-Wallis test compared with the $F$-test, and for testing the location parameter of a single symmetric distribution. A number of alternative notions of asymptotic efficiency are compared in Section 2. In this connection, certain difficulties arise because power is not necessarily a convex function of sample size. As an alternative to the Pitman notion of asymptotic efficiency, we consider in Section 3 one based on the speed with which power at a fixed alternative tends to 1. In particular we obtain, for the sign test relative to the $t$ in normal populations, the limit as $n \rightarrow \infty$ of the sequence of power efficiency functions. It is noted that certain interchanges of limit passages are not always possible.

This study was initiated in connection with estimating parameters involved in a certain stochastic process of population growth. Because of the nature of distribution functions arising in such studies, the usual methods of estimation result in formulas which are so complex that it is difficult, if not impossible, to obtain explicit solutions for the estimates of the parameters. Investigation of the problem led to an extension of the method of best asymptotically normal estimates developed by Neyman [1]. The estimates derived are termed regular best asymptotically normal estimates (RBAN estimates). This extension can be applied to other problems. In [1], Neyman considers a whole class of estimates which possess the properties of consistency, of asymptotic normality, and of asymptotic efficiency, and he provides estimates having these asymptotic properties for the case of multinomial distributions. His method is extended in the present paper to a more general case in which random vectors are dealt with. Such an extension was considered by Barankin and Gurland [2], who studied a large class of estimates and showed that if the distributions involved are members of Koopman's family, it is still possible to reach the Cramer-Rao lower bound. The purposes of the present paper are to discuss a subclass of the estimates considered by Barankin and Gurland and to present simple methods of generating such estimates. The estimates discussed are based on a number of independent random vectors whose distribution functions are not specified. It is proved that under certain regularity conditions, the regular and consistent estimates obtained are asymptotically normal as the number of random vectors tends to infinity. A necessary and sufficient condition for a regular and consistent estimate to have a "minimal" asymptotic covariance matrix is given. An expression is derived for the "minimal" asymptotic covariance matrix. It is also proved that if a function $\mathbf{f}$ satisfies certain conditions, then in order that $\mathbf{f}(\tilde\theta)$ be an RBAN estimate of $\mathbf{f}(\theta)$ at $\mathbf{f}(\theta^0)$, where $\theta^0$ is the true value of the parameter point $\theta$, it is necessary and sufficient that the argument $\tilde\theta$ be an RBAN estimate of $\theta$ at $\theta^0$. Methods of generating RBAN estimates are given. For simplicity of presentation, matrix notation is used throughout this paper. By derivatives of a matrix with respect to a vector (or with respect to a second matrix) is meant the derivatives of the matrix simultaneously with respect to all the components of the vector (or all the elements of the second matrix). The usual rules of differentiation with respect to vectors are used.

This paper studies the large-sample power of certain rank order tests against one-parameter alternatives in the two-sample problem. The first $m$ of $N$ independent random variables are supposed identically distributed, each with a density function $f_1(x, \theta)$, the remaining $N - m$ with a density function $f_2(x, \theta)$. When $\theta = 0$ both density functions are the same. Let $a_{N1}, \cdots, a_{NN}$ be a set of constants defined by (3.2) below; let $b_{N1}, \cdots, b_{NN}$ be another set of constants; and let $R_1, \cdots R_N$ be the ranks of the $N$ random variables. A statistic of the type $\sum^N_{i = 1} a_{Ni}b_{NR_i}$ is called an $L$ statistic. Part I of this paper characterizes the locally best rank order statistic for testing $H_0:\theta = 0$ against the alternative that $\theta$ is positive and "close" to zero. This turns out to be any one of an equivalent class of $L$ statistics. Under certain regularity conditions it is possible to determine the large-sample power of $L$ statistics. Of particular interest is the large-sample power of the locally best $L$ statistic. For arbitrary $b_{N1}, \cdots, b_{NN}$ it is usually difficult to determine whether the regularity conditions hold. Hence, in Part II a special class of $L$ statistics, the $L_h$ statistics, are studied. For these, the regularity conditions are easier to verify and the large-sample power is determined. The best $L$ statistic can, in a certain sense, be approximated by $L_h$ statistics.

The estimation of the size of an animal population is considered for a situation where the population is stratified and only partial mixing takes place between strata. A consistent estimate is found and its variance determined. It is shown that estimates previously given or frequently used in this situation are not necessarily consistent and, in fact, may be meaningless. Conditions for their consistency are determined. Some further statistical problems in estimating the interstrata migration are discussed.

It may frequently happen that a researcher, wishing to decide which one of a set of alternatives to accept, finds that there are several experiments available to him which he might perform to guide him in reaching his decision. Thus he is faced with making a preliminary decision as to which experiment or experiments he is to perform. If he admits the possibility of performing more than one, then the question of how many, which ones, and in what order arises. It is such questions as these that come under the heading of comparison and design of experiments. While a great deal of general theory of the design problem has been developed, e.g., by Wald [1] and Maguire [2], few actual solutions of particular problems, especially of the sequential type, have been investigated thus far. Robbin's paper [3] is the first published report dealing with various sequential rules for particular nontruncated design problems. The basic purpose of this paper is to investigate for certain cases the optimal design, which almost uniformly turns out to be exceedingly complicated (see Section 3), and to propose and determine some justification for certain simpler criteria. Attention is restricted to problems in which there are but two alternatives, or hypotheses, $H_1$ and $H_2$, and it is required to decide between them with a loss of one unit if the false one is selected, while no loss results from selecting the correct one. Further, $\xi$ will denote the a priori probability that $H_1$ is true, and the basic criterion for comparison will be the Bayes risks associated with the various experiments. To say that an experiment is available to a researcher is to say that there is a real random variable which he can observe whose distribution is known under each hypothesis. As an example of a situation in which this type of question may arise, consider the problem of deciding between utilizing a use test as against a specifications test for acceptance of a lot of manufactured items. A large lot of items has been produced and a decision is to be made between, say, $w_1$ and $w_2$ as being the proportion of defectives in the lot. Let $X = 1$ or 0 according as an item selected at random is defective or not as determined by subjecting it to a use test. Let $Y = 1$ or 0 according as an item selected at random is classified as defective (because it fails to meet certain specifications) or not. If $\alpha$, the probability that a nondefective item fails to meet the specifications, and $\beta$, the probability that a defective item meets the specifications, are known, then both $X$ and $Y$ have a binomial distribution with known parameter under each hypothesis. Again, it might be that in the course of a series of treatments of a material there are two points at which a certain characteristic may be measured, say the breaking strength of a metal undergoing a series of heat treatments. Let $X$ and $Y$, respectively, denote the value of the characteristic at the different points in the process. It is reasonable to assume that under each of two simple hypotheses concerning the process and material, $X$ and $Y$ have prescribed normal distributions. In general, suppose that $X$ and $Y$ are two real random variables having distribution functions of $F_i$ and $G_i$, respectively, under hypothesis $H_i$ and with corresponding densities $f_i$ and $g_i$ with respect to a common measure, $\Psi$, such that $f_i > 0$ if and only if $g_i > 0$. Let $R_Z(\xi)$ denote the Bayes risk against $\xi$ when using experiment $Z$. Now the computation and comparison of the risk, $R_X(\xi)$ and $R_Y(\xi)$, for all $\xi$, which appears to be necessary in order to obtain an optimal design, is intrinsically complicated in most cases, as will be seen in the following sections. Hence it is of some interest to investigate some more convenient criteria for choosing between experiments. Any such criterion should, of course, dictate the use of $X$ if $R_X \leqq R_Y$ (i.e., $R_X(\xi) \leqq R_Y(\xi)$ for all $\xi$). To be able to check on this is but one reason for interest in conditions that $R_X \leqq R_Y$. Another is that whenever it is true that $R_X \leqq R_Y$, then regardless of the choice of actions open to the researcher or the loss function used, use of $X$ will never yield a greater Bayes risk than $Y$. Also, if a total of, say, $n$ independent experiments is to be performed, the optimal sequential design is the nonsequential rule: take all $n$ observations of $X$ ([4], [5], [8]). In Section 2.1 general conditions that $R_X \leqq R_Y$ are derived, some of which are related to those obtained by Blackwell [5] via consideration of the standard experiment. In Section 2.2 the Kullbach-Leibler (abbreviated hereafter as K-L) information numbers are introduced, and it is shown, in particular, that they provide a criterion which yields an especially simple necessary condition that $R_X \leqq R_Y$. The K-L information numbers are also considered as functions of that transformation, $t$, such that the distribution of $t(X)$ under $H_1$ is the distribution of $X$ under $H_2$. The case in which all the distributions involved are normal is analyzed in some detail in Section 2.3, where it is seen that the K-L numbers do not yield a sufficient condition that $R_X \leqq R_Y$, though they do yield a sufficient condition that $R_X = R_Y$. The normal case gives an example in which a second criterion, that of being "locally more informative" (Bayes) at both zero and 1, yields a condition both necessary and sufficient that $R_X \leqq R_Y. X$ is termed locally more informative than $Y$ at $\xi$ if $\xi$ lies in an interval $\lbrack\xi_1, \xi_2\rbrack$ such that on $\lbrack 0,1\rbrack \cap \lbrack\xi_1, \xi_2\rbrack R_X(\xi) \leqq R_Y(\xi)$, with strict inequality at at least one end point. This latter criterion is discussed further in Section 2.4. The problem of determining the optimal designs, sequential and nonsequential, for the case in which all distributions are binomial and a fixed number of experiments is to be performed is discussed in Section 3. The complete solutions are found to be exceedingly complicated; a few are given. For the sequential design, a system for obtaining the optimal design which avoids the complete calculation of the successive risk functions was found. In the final section, a sequential rule for terminating experimentation is considered, and the problem of finding a sequential design which minimizes the expected number of experiments is posed. Two reasonable designs are proposed, are shown to be equivalent, and are shown to be better than either of the rules which require that all observations be of the same random variable. Throughout the paper it will be convenient to consider $\xi/(1 - \xi)$, and this will regularly be denoted by $\eta$.

Tables of the means, variances, and covariances, to five decimal places, of order statistics from samples of size ten or less have been given by Godwin [3]. In this paper basic expected values are given of order statistics and products of order statistics, for samples of size twenty and less to 10 decimal places (DP). In addition, certain other functions are tabulated to 25 DP to facilitate extension to larger sample sizes.

Waiting-line or queuing processes of the Markov type are studied, the incoming traffic being of Poisson type and having negative-exponential holding time. The parameters are allowed to depend on time. The problem of finding an exact solution for the probability distribution of the waiting-line length as a function of time is reduced to the solution of an integral equation of the Volterra type. When the ratio of the parameters for the incoming and out-going traffic is constant, this equation can be solved explicitly and the required distribution obtained. Using this solution, the behavior of the process for large values of $t$ is studied, particularly for the unstable case with traffic intensity $\geqq 1$.

In this paper, methods introduced earlier by the author [1] are used to obtain simple, accurate formulas for the decision boundaries for sequential probability ratio tests for simple hypotheses and alternatives on the mean $\theta$ of the exponential distribution $\theta^{-1} \exp(-u/\theta)$. Examples are provided to indicate the accuracy and the degree of complexity of the results. It is hoped that the results given here will find applications in life testing and statistical studies of radioactive decay.

In this paper a new normal approximation to a sum of hypergeometric terms is derived, which is a direct generalization of Feller's normal approximation to the binomial distribution [2]. For intervals that are asymmetric with respect to the mean, or when the distribution is skewed, the new approximation is a marked improvement over the classical procedure. The hypergeometric distribution is discussed in Section 2, along with the classical norming and the resulting approximation. Feller's remarkable normal approximation for the related binomial distribution is given in Section 3 with an indication of how it can be extended to cover the hypergeometric case. The result of such an extension is presented in Theorem 2 of Section 4. This theorem gives upper and lower bounds on the hypergeometric sum and hence provides a useful estimate of the relative error. Preliminary results to proving Theorem 2 are exhibited in Section 5. The proof follows in Section 6.

A table is given to simplify the estimation of the parameters of an incomplete gamma or Type III distribution. A new procedure is also suggested for estimating the parameters of a truncated gamma distribution. This method is also applicable to a number of other truncated distributions, whether the truncation is in the tails or the center of the distribution.

Given a $k$-fold multinomial distribution with equal probability for each category, the probability of the largest frequency in any category is desired. A simple asymptotic approximation to the upper percentage points of this distribution is obtained. A table of .95 and .99 points of the approximation for $k = 1(1)25$, and a table comparing these with actual values for $k = 3, 4, 5$ and $n = 3(1)12$, are provided. An investigation of the moment problem is given.

Let $x_1, x_2, \cdots, x_n, x'_1, x'_2, \cdots, x'_{nk}$ be independent random variables with a common continuous distribution $F(x)$. Let $x_1, x_2, \cdots, x_n$ have the empiric distribution $F_n(x)$ and $x'_1, x'_2, \cdots, x'_{kn}$ have the empiric distribution $G_{nk}(x)$. The exact values of $P(-y < F_n(s) - G_{nk}(s) < x$ for all $s$) and $P(-y < F(s) - F_n(s) < x$ for all $s$) are obtained, as well as the first two terms of the asymptotic series for large $n$.

In this paper we have solved certain distribution problems connected with the simultaneous analysis of variance test [1] and have proved that the power function of this test has the monotonicity property.

In an investigation of the distribution of the likelihood ratio $\lambda$, Wilks [3] proved, under certain regularity conditions, that $-2 \ln \lambda$ is, except for terms of order $1/\sqrt n$, distributed like $\chi^2$ with $k - m$ degrees of freedom, where $k$ is the dimension of the parameter space $\Omega$ of admissible hypotheses and $m$ is the dimension of the parameter space $\omega$ of null hypotheses. In this paper, we consider the nonregular densities investigated by R. C. Davis [1] and show that for certain hypotheses $-2 \ln \lambda$ has an exact $\chi^2$-distribution with $2(k - m)$ degrees of freedom.

Let $M(x)$ be a strictly increasing regression function for $x < \theta$, and strictly decreasing regression function for $x > \theta$. Under conditions 1, 2, and 3 given below, the stochastic approximation procedure proposed by Kiefer and Wolfowitz [3] is shown to converge stochastically to $\theta$. Under the additional conditions 4, 5, 6 given below, the procedure is shown to converge in distribution to the normal distribution. Our method is the one used by Chung [2].

Hotelling and Pabst [1] showed that the rank correlation coefficient had a limiting normal distribution under the equally likely permutations of the hypothesis of independence. Wald and Wolfowitz [2] developed a general theorem of this type, and Noether [3] and Hoeffding [4] have relaxed the conditions used therein. In this paper a vector form of the theorem is proved along the lines used in an example by Wald and Wolfowitz [1] but taking account of the singular cases in which the correlations approach one.