An infinite class of decision rules having several desirable properties is suggested for choosing a group of populations from a given set of normal populations which should contain the population with the largest mean. The problem of selecting one member from this infinite class of rules has also been studied.

A comparison is made of several definitions of ordered sets of distributions, some of which were introduced earlier by the author [7], [8] and by Rubin [10]. These definitions attempt to make precise the intuitive notion that large values of the parameter which labels the distributions go together with large values of the random variables themselves. Of the various definitions discussed the combination of two, (B) and (C) of Section 2, appears to be statistically most meaningful. In Section 3 it is shown that this ordering implies monotonicity for the power function of sequential probability ratio tests. In Section 4 the results are applied to obtaining tests that give a certain guaranteed power with a minimum number of observations. Finally, in Section 5, certain consequences are derived regarding the comparability of experiments in the sense of Blackwell [1].

By a statistical design (or simply, a design) we mean an arrangement of a certain number of "treatments" in a certain number of "blocks" in such a way that some prescribed combinatorial conditions are fulfilled. With every design is associated a unique matrix called the incidence matrix of the design (definitions, etc., in subsequent sections). In many instances, e.g., [7] [8], [10], [12], [16], information regarding certain kinds of designs such as BIB, PBIB designs is obtained from properties of the matrix $NN'$ or of its determinant $|NN'|$ where $N$ is the incidence matrix of the design under consideration. On the other hand in a few cases, such as [4], [5], [11], [14], [15], the incidence matrix $N$ itself has been to investigate properties of designs. This paper gives a method of using incidence matrices of known designs to obtain new designs. In Section 2 we have defined the Kronecker product of matrices. This definition and some properties of the Kronecker product of matrices are given in [1]. Section 3 is devoted to a general discussion of an application of the concept of the Kronecker product of matrices to define the Krnoecker product of designs. This section also contains two theorems which illustrate the use of the method of obtaining Kronecker products of designs. Definitions of some well-known designs are given in Section 4, which also contains a number of results giving explicit forms of certain Kronecker products. Finally some illustrations of a few results of Section 4 are given in Section 5.

The problem is considered of obtaining bounds for the (cumulative) distribution function of the sum of $n$ independent, identically distributed random variables with $k$ prescribed moments and given ranger. For $n = 2$ it is shown that the best bounds are attained or arbitrarily closely approach with discrete random varibles which take on at most $2k + 2$ values. For nonnegative random variables with given mean, explicit bounds are obtained when $n = 2$; for arbitrary values of $n$, bounds are given which are asymptotically best in the "tail" of the distribution. Some of the results contribute to the more general problem of obtaining bounds for the expected values of a given function of independent, identically distributed random variables when the expected values of certain functions of the individual variables are given. Although the results are modest in scope, the authors hope that the paper will draw attention to a problem of both mathematical and statistical interest.

Some invariant procedures, which are essentially step-functions, are considered as estimators of the cumulative distribution function of a one-dimensional random variable on which a finite fixed number of observations are given, for various loss functions. Two principal classes of loss functions are considered and it is shown that for a special loss function in one class the optimum procedure is the usual sample cumulative function.

The authors were prompted by a general problem concerning hit probabilities arising in military operations to seek the distribution of $Q_i = \sum^k_{i=1}a_ix^2_i, k = 2, 3,$ where the $x_i$ are normally and independently distributed with zero mean and unit variance, $\sum a_i = 1,$ and $a_i > 0.$ While the distribution of a positive definite quadratic form in independent normal variates has been the subject of several papers in recent years [6], [11], [12], laborious computations are required to prepare from existing results the percentiles of the distribution and a table of hit probabilities. This paper discusses the exact distribution of $Q_k$ and then obtains and tabulates the distributions of $Q_2$ and $Q_3,$ accurate to four places. Three other approaches to the distributions are discussed and compared with the exact results: a derivation by Hotelling [8], the Cornish-Fisher asymptotic approximation [3], and the approximation obtained by replacing the quadratic form with a chi-square variate whose first two moments are equated to those of the quadratic form--a type of approximation used in components of variance analysis. The exact values and the approximations are given in Tables I and II. The tables have been prepared with the original problem in mind, but also serve as an aid in several problems arising out of quite different contexts, [1], [2], [13]. These are discussed in Section 6.

In a special case of Polya's urn scheme, the probability that the first $n$ draws are all of the same color is interpreted as a function of the (single) correlation coefficient. A more general urn model is introduced in which the correlation between pairs of results may differ from pair to pair, and again the probability of consecutive colors is considered. This result is compared with the probability of coincidence in sign under the multivariate normal distribution. The comparison suggests a new approximation for the probability in the multivariate normal case. This approximation appears to be useful only in the Polya case, where the correlations are all equal.

Let $(X, S, \mu)$ be a probability measure space: $X$ is set of points $x, S$ is a field of subsets of $X,$ and $\mu$ is a countably additive measure on $S$ with $\mu(X) = 1.^2$ A subfield is a field $S_0$ of subsets of $X$ such that $S_0 \subseteq S,$ that is, each $S_0$-measurable set is also $S$-measurable. A statistic is a function defined on $X.$ There is no a priori restriction on the class of statistics; in particular, statistics are not necessarily real-valued, and a real-valued statistic is not necessarily an $S$-measurable function. For any statistic $f,$ let $S_f$ denote the class of all sets which are $S$-measurable and of the form $f^{-1}(B),$ where $B$ is a subset of the range of $f.$ The class $S_f$ is clearly a subfield, and is called the subfield induced by $f.$ The induced subfield $S_f$ plays a central role in the study of a statistic $f,$ for the following reason. The probabilist or mathematical statistician is usually concerned not with the statistic $f$ as such, but rather with the class of random variables (i.e., real-valued $S$-measurable functions) which depend on $x$ only through $f,$ and, as is easily seen, this class of random variables is exactly the class of real-valued $S_f$-measurable functions. In case the given statistic $f$ is a random variable (and therefore itself an object of study), the argument just given continues to apply, because in this case $f$ is necessarily an $S_f$-measurable function. This paper discusses certain measure-theoretic problems concerning the relations between subfields, subfields of the apparently special form $S_f,$ and statistics. The main problems, as also the main conclusions, are discribed in the following paragraphs. Most of the conclusions of the paper are valid only in the case when $(X, S)$ is (or may be taken to be) a euclidean sample space, that is, $X$ is a Borel set of the $m$-dimensional euclidean space $(1 \leqq m \leqq \infty),$ and $S$ is the field of Borel sets of $X.$ It is assumed henceforth that this is the case under consideration. There are two main problems. The first is whether every subfield is inducible by a statistic. This problem is discussed (in a more general setting) in [2], and the conclusions of the present paper complement those of [2]. It is shown here that every subfield is inducible by a statistic if and only if the sample space is discrete, that is to say, $X$ is a countable set and $S$ is the class of all subsets of $X$ (Theorem 1). This result is, however, not quite relevant to situations where the natural equivalence relation between subfields is not identity but approximability to within sets of $\mu$-measure zero. The equivalence relation referred to is defined as follows. A subfield $S_1$ is a contraction of a subfield $S_2$ if corresponding to each real-valued $S_1$-measurable function $f_1$ there exists an $S_2$-measurable function $f_2$ such that $f_2(x) = f_1(x)$ except on a set of $\mu$-measure zero; we then write $S_1 \subseteq S_2 \lbrack S, \mu\rbrack.$ The subfields $S_1$ and $S_2$ are equivalent if each is a contraction of the other; we then write $S_1 = S_2 \brack S, \mu\rbrack.$ It is shown that, in fact, corresponding to any subfield $S_0$ there exists an $f$ such that $S_f$ is equivalent to $S_0,$ and that this $f$ may be taken to be a random variable (Theorem 2). In the literature the notion of contraction (and the derived notion is equivalence) has been defined for statistics in two ways, which are here called contraction and functional contraction. A statistic $f$ is a contraction of a statistic $g$ if $S_f$ is a contraction of $S_g$ (that is, $S_f \subseteq S_g \lbrack S, \mu\rbrack ); f$ is a functional contraction of $g$ (written $f \subseteq g \lbrack S, \mu\rbrack $) if there exists a function $h$ on the range of $g$ into that of $f,$ and an $S$-measurable set $N$ with $\mu(N) = 0,$ such that $f(x) = h(g(x))$ for $x$ in $X - N.$ (Cf. [3], [4].) It seems to the write that for most (possibly all) technical purposes the relevant concept is contraction as just defined (cf. Lemmas 7.1 and 3.2 of [1]). However, functional contraction has simpler interpretations and greater intuitive appeal. The second problem is the exact relation between contraction and functional contraction. It is shown that, in general, functional contraction does not imply contraction (Example 1), and also that contraction does not imply functional contraction (Example 2). If, however, both $f$ and $g$ are random variables, then $S_f \subseteq S_g [S, \mu]$ if and only if $f \subseteq g [S, \mu]$ (Theorem 3). It follows, in particular, that if the sample space is discrete, then contraction coincides with functional contraction. The problems described above arose in connection with the theory of sufficiency, and the results have applications in that theory. It follows, for example (assuming that the sample space is euclidean and that the set of alternative distributions of the sample point is a dominated set),that it $f$ is a necessary and sufficient statistic, then $S_f$ is a necessary and sufficient subfield (Corollary 2). The following are some general conclusions bearing on mathematical models for studies such as [1]. (a) The notion of a subfield, while certainly no loss general than that of a statistic, is in fact no more general. (b) There is no loss of generality, or other disadvantage, in defining a statistic to be a random variable. On the contrary, admission of nonmeasurable functions to the discussion leads to inconsistencies between extension and functional extension--this seems undesirable. (c) If $f$ is a random variable, it is inmaterial whether $f$ is regarded as a statistic or as a Borel-measurable transformation (cf. [1], p. 431). These satisfactory conclusions do not necessarily hold for an arbitrary space $(X, S, \mu).$ An example given in [2] shows that at least (a) and (c) are not valid in the general case.

This paper gives maximum likelihood estimators of parameters of truncated and censored exponential distributions, asymptotic variances of the estimators, and asymptotic confidence intervals for the parameters. Applications to bombing accuracy studies and to life testing are pointed out. As regards bombing accuracy the parameter estimated is the reciprocal of the variance in a normal bivariate distribution having circular symmetry. The reciprocal is estimated because there is no maximum likelihood estimator of the variance and any estimator of the variance is badly biased (see Section 2). Results of a synthetic sampling experiment are given to provide information on rapidity of convergence of the distributions of the estimators to their asymptotic distributions.

Consider a finite set $\Omega$ of $N$ points and a single-valued function $f(x)$ on $\Omega$ into $\Omega.$ In case the mapping is one-to-one, it is a permutation of the points of $\Omega;$ we shall be concerned with more general mappings. Any mapping function effects a decomposition of the set into disjoint, minimal, non-null invariant subsets, as $\Omega = \omega_1 + \omega_2 + \cdots + \omega_k,$ where $f(\omega_i) \subset \omega_i$ and $f^{-1}(\omega_i) \subset \omega_i.$These subsets have been referred to as trees and as components of the mapping; we shall say that $f,$ as above, decomposes the set into $k$ components. Metropolis and Ulam [1] defined a random mapping by a uniform probability distribution over the $\Omega^\Omega$ sample points of $f(x)$ and posed the problem of finding the expected number of components. Kruskal [2] subsequently solved this problem. In this paper, we consider a related problem, namely, what is the probability that a random mapping is indecomposable, i.e., that the minimal non-null set $\omega$ for which $f(\omega) = \omega$ and $f^{-1}(\omega) = \omega,$ is the whole set $\omega = \Omega?$ This problem is solved in general, as is, also, an analogous problem for a specialized random mapping of some interest in social psychology. Finally, we examine the asymptotic behavior of these probabilities.

In Section 2 we have defined the Kronecker product of matrices. This definition and some properties of the Kronecker product of matrices are given in [1]. Section 3 is devoted to a general discussion of an application of the concept of the Kronecker product of matrices to define the Krnoecker product of designs. This section also contains two theorems which illustrate the use of the method of obtaining Kronecker products of designs. Definitions of some well-known designs are given in Section 4, which also contains a number of results giving explicit forms of certain Kronecker products. Finally some illustrations of a few results of Section 4 are given in Section 5.

Let $\mu$ be a probability measure on the Borel sets of $k$-dimensional Euclidean space $E_k.$ Let ${X_n}, n = 1, 2, \cdots,$ be a sequence of $k$-dimensional independent random vectors, distributed according to $\mu.$ For each $n = 1, 2, \cdots$ let $\mu_n$ be the empiric distribution function corresponding to $X_1, \cdots, X_n,$ i.e., for every Borel set $A \epsilon E_k,$ we define $\mu_n(A)$ to be the proportion of observations among $X_1, \cdots, X_n$ which fall in $A.$ Let $\alpha$ be the class of Borel sets in $E_k$ defined below. The object of this paper is to prove that $P{\lim_{n\rightarrow\infty}} \sup_{A \epsilon \mathscr{a} \|\mu_n(A) - \mu(A)\| = 0} = 1.$

Let $f(x)$ be a bounded density function over the finite interval [A, B] with at most a finite number of discountinities. Let $X_1, X_2, \cdots, X_n$ be independent chance variables each with the density $f(x).$ Define $Y_1 \leqq Y_2 \leqq \cdots \leqq Y _n$ as the ordered values of $X_1, X_2, \cdots, X_n,$ and $T_i$ as $Y_{i+1} - Y_i.$ Also define $R_n(t)$ as the proportion of the variates $T_1, \cdots, T_{n-1}$ not greater than $t / (n - 1).$ We shall denote $\lbrack 1 - \int^B_A fxe^{-tf(x)} dx=\rbrack$ by $S(t),$ and $\sup_{t\geqq 0} \|R_n(t) - S(t)\|$ by $V(n).$ Then it is shown that as $n$ increases, $V(n)$ converges stochastically to zero. The relation of this result to other results is discussed.