The foundations of a general theory of statistical decision functions, including the classical non-sequential case as well as the sequential case, was discussed by the author in a previous publication [3]. Several assumptions made in [3] appear, however, to be unnecessarily restrictive (see conditions 1-7, pp. 297 in [3]). These assumptions, moreover, are not always fulfilled for statistical problems in their conventional form. In this paper the main results of [3], as well as several new results, are obtained from a considerably weaker set of conditions which are fulfilled for most of the statistical problems treated in the literature. It seemed necessary to abandon most of the methods of proofs used in [3] (particularly those in section 4 of [3]) and to develop the theory from the beginning. To make the present paper self-contained, the basic definitions already given in [3] are briefly restated in section 2.1. In [3] it is postulated (see Condition 3, p. 207) that the space $\Omega$ of all admissible distribution functions $F$ is compact. In problems where the distribution function $F$ is known except for the values of a finite number of parameters, i.e., where $\Omega$ is a parametric class of distribution functions, the compactness condition will usually not be fulfilled if no restrictions are imposed on the possible values of the parameters. For example, if $\Omega$ is the class of all univariate normal distributions with unit variance, $\Omega$ is not compact. It is true that by restricting the parameter space to a bounded and closed subset of the unrestricted space, compactness of $\Omega$ will usually be attained. Since such a restriction of the parameter space can frequently be made in applied problems, the condition of compactness may not be too restrictive from the point of view of practical applications. Nevertheless, it seems highly desirable from the theoretical point of view to eliminate or to weaken the condition of compactness of $\Omega$. This is done in the present paper. The compactness condition is completely omitted in the discrete case (Theorems 2.1-2.5), and replaced by the condition of separability of $\Omega$ in the continuous case (Theorems 3.1-3.4). The latter condition is fulfilled in most of the conventional statistical problems. Another restriction postulated in [3] (Condition 4, p. 297) is the continuity of the weight function $W(F, d)$ in $F$. As explained in section 2.1 of the present paper, the value of $W(F, d)$ is interpreted as the loss suffered when $F$ happens to be the true distribution of the chance variables under consideration and the decision $d$ is made by the statistician. While the assumption of continuity of $W(F, d)$ in $F$ may seem reasonable from the point of view of practical application, it is rather undesirable from the theoretical point of view for the following reasons. It is of considerable theoretical interest to consider simplified weight functions $W(F, d)$ which can take only the values 0 and 1 (the value 0 corresponds to a correct decision, and the value 1 to a wrong decision). Frequently, such weight functions are necessarily discontinuous. Consider, for example, the problem of testing the hypothesis $H$ that the mean $\theta$ of a normally distributed chance variable $X$ with unit variance is equal to zero. Let $d_1$ denote the decision to accept $H$, and $d_2$ the decision to reject $H$. Assigning the value zero to the weight $W$ whenever a correct decision is made, and the value 1 whenever a wrong decision is made, we have: $W(\theta, d_1) = 0 \text{for} \theta = 0, \text{and} = 1 \text{for} \theta \neq 0; W (\theta, d_2) = 0 \text{for} \theta \neq 0, \text{and} = 1 \text{for} \theta = 0.$ This weight function is obviously discontinuous. In the present paper the main results (Theorems 2.1-2.5 and Theorems 3.1-3.4) are obtained without making any continuity assumption regarding $W(F, d)$. The restrictions imposed in the present paper on the cost function of experimentation are considerably weaker than those formulated in [3]. Condition 5 [3, p. 297] concerning the class $\Omega$ of admissible distribution functions, and condition 7 [3, p. 298] concerning the class of decision functions at the disposal of the statistician are omitted here altogether. One of the new results obtained here is the establishment of the existence of so called minimax solutions under rather weak conditions (Theorems 2.3 and 3.2). This result is a simple consequence of two lemmas (Lemmas 2.4 and 3.3) which seem to be of interest in themselves. The present paper consists of three sections. In the first section several theorems are given concerning zero sum two person games which go somewhat beyond previously published results. The results in section 1 are then applied to statistical decision functions in sections 2 and 3. Section 2 treats the case of discrete chance variables, while section 3 deals with the continuous case. The two cases have been treated separately, since the author was not able to find any simple and convenient way of combining them into a single more general theory.

The multiplicative process is usually defined by the sequence of random variables $X_0, X_1, \cdots$ whose distributions are specified as follows: $P(X_0 = 1) = 1, \sum^\infty_{\nu=0} P(X_1 = \nu) = 1,$ and if $X_n = 0$ then $P(X_{n+1} = 0) = 1,$ whereas if $X_n$ is a positive integer then $X_{n+1}$ is distributed as the sum of $X_n$ independent random variables each with the distribution of $X_1$. The variable $X_n$ is interpreted as the number of "particles" in the $n$th generation, and the index $n$ as a discrete time parameter. This has been the method of approach in previous studies of the process [1, 2, 3, 4, 5]. The multiplicative process has various applications, notably in the study of population growth, the spread of epidemics of rumors, and the nuclear chain reaction. The closely related "birth and death" process was recently studied by Kendall [6]. Whenever one studies the probability theory of a particular system there seem to be definite conceptual advantages in defining explicitly the set $\mathcal{J}$ of elementary events, the additive class $\mathcal{M}$ of subsets of $\mathcal{J}$, called events, and the probability measure $P$ for the events of $\mathcal{M}$. Now an elementary event of this process can be represented by a rooted tree where the original particle is represented by the root vertex and where the particles of the $n$th generation are represented by the vertices $n$ segments removed from the root. The tree will be finite or infinite according to whether a finite or an infinite number of particles are involved in the elementary event. Thus, the set of trees is the natural choice for $\mathcal{J}$. The first part of this paper is devoted to a more precise description of $\mathcal{J, M}$ and $P$. We shall then see easily that $X_n(t)$, the number of vertices $n$ segments removed from the root of $t \epsilon \mathcal{J}$, i.e. the number of particles in the $n$th generation, has the distribution defined in the preceding paragraph. Since the time does not appear in our description of $\mathcal{J}$ we fetter ourselves somewhat if we interpret $n$ as a discrete time parameter. Thus, we have already reaped some harvest from considering the process from the point of view of $\mathcal{J}$. Another advantage is that we are led in a natural way to study the distribution of other structural features of the trees, e.g. the total number of vertices, or the number of vertices with $k$ outgoing segments. The chief results of this paper are as follows. The recursion formula for the probability $P_n$ that a tree have $n$ vertices $n = 1, 2, \cdots$ is obtained as well as an asymptotic estimate of $P_n$ valid for large $n$. The distributions of the number of branches at the root in a finite tree, an infinite tree, or in a tree with $n$ vertices are obtained and the asymptotic distribution of the latter as $n \rightarrow \infty$. The distribution of the fraction of vertices with $k$ outgoing segments in the finite trees, in the trees with $n$ vertices, and the asymptotic distribution of the latter as $n \rightarrow \infty$ are also found. Finally, an estimate is obtained for the probability that a tree be finite in case this probability is near 1, a result which was previously obtained by Kolmogoroff [7].

The body of this paper is written in terms of very general and abstract ideas which have been popular in pure mathematical work on the theory of probability for the last two or three decades. It seems to us that these ideas, so fruitful in pure mathematics, have something to contribute to mathematical statistics also, and this paper is an attempt to illustrate the sort of contribution we have in mind. The purpose of generality here is not to solve immediate practical problems, but rather to capture the logical essence of an important concept (sufficient statistic), and in particular to disentangle that concept from such ideas as Euclidean space, dimensionality, partial differentiation, and the distinction between continuous and discrete distributions, which seem to us extraneous. In accordance with these principles the center of the stage is occupied by a completely abstract sample space--that is a set $X$ of objects $x$, to be thought of as possible outcomes of an experimental program, distributed according to an unknown one of a certain set of probability measures. Perhaps the most familiar concrete example in statistics is the one in which $X$ is $n$ dimensional Cartesian space, the points of which represent $n$ independent observations of a normally distributed random variable with unknown parameters, and in which the probability measures considered are those induced by the various common normal distributions of the individual observations. A statistic is defined, as usual, to be a function $T$ of the outcome, whose values, however, are not necessarily real numbers but may themselves be abstract entities. Thus, in the concrete example, the entire set of $n$ observations, or, less trivially, the sequence of all sample moments about the origin are statistics with values in an $n$ dimensional and in an infinite dimensional space respectively. Another illuminating and very general example of a statistic may be obtained as follows. Suppose that the outcomes of two not necessarily statistically independent programs are thought of as one united outcome--then the outcome $T$ of the first program alone is a statistic relative to the united program. A technical measure theoretic result, known as the Radon-Nikodym theorem, is important in the study of statistics such as $T$. It is, for example, essential to the very definition of the basic concept of conditional probability of a subset $E$ of $X$ given a value $y$ of $T$. The statistic $T$ is called sufficient for the given set $\mathcal{M}$ of probability measures if (somewhat loosely speaking) the conditional probability of a subset $E$ of $X$ given a value $y$ of $T$ is the same for every probability measure in $\mathcal{M}$. It is, for instance, well known that the sample mean and variance together form a sufficient statistic for the measures described in the concrete example. The theory of sufficiency is in an especially satisfactory state for the case in which the set $\mathcal{M}$ of probability measures satisfies a certain condition described by the technical term dominated. A set $\mathcal{M}$ of probability measures is called dominated if each measure in the set may be expressed as the indefinite integral of a density function with respect to a fixed measure which is not itself necessarily in the set. It is easy to verify that both classical extremes, commonly referred to as the discrete and continuous cases, are dominated. One possible formulation of the principal result concerning sufficiency for dominated sets is a direct generalization to the abstract case of the well known Fisher-Neyman result: $T$ is sufficient if and only if the densities can be written as products of two factors, the first of which depends on the outcome through $T$ only and the second of which is independent of the unknown measure. Another way of phrasing this result is to say that $T$ is sufficient if and only if the likelihood ratio of every pair of measures in $\mathcal{M}$ depends on the outcome through $T$ only. The latter formulation makes sense even in the not necessarily dominated case but unfortunately it is not true in that case. The situation can be patched up somewhat by introducing a weaker notion called pairwise sufficiency. In ordinary statistical parlance one often speaks of a statistic sufficient for some of several parameters. The abstract results mentioned above can undoubtedly be extended to treat this concept.

In designing single sampling inspection plans, a problem is to find the acceptance number, $c$, and the smallest sample size, $n$, such that if the fraction defective of the material inspected is equal to an acceptable value, $p_1$, a large percentage, say, 95% of such lots will be accepted under the sample criteria, whereas if the fraction defective of the material inspected is objectionable and equal to $p_2$ (where $p_1 < p_2$), then a large percentage, say, 90% of such lots will be rejected. A solution to this problem for the case where the lot size is large compared to the sample size is given in this paper and tables are provided for quick determination of the sample size $n$ and acceptance number $c$.

This paper is divided into two parts. The significance tests investigated in Part I concern the population mean and are based on the quantity [(sample midrange)-(hypothetical mean)]/(sample range). The case in which the observations are a sample from a normal population is considered in detail. The tests investigated are summarized in Table 1. These tests are found to be very efficient for small samples (see Table 4, power efficiency is defined in section 3). An investigation of several extremely non-normal populations using the values of $D\alpha$ obtained for normality indicates that the significance level of the range-midrange test is not very sensitive to the requirement of normality for small samples (see Table 6). Also the tests of Table 1 can be applied without computation through the use of an easily constructed graph (see section 4). These properties suggest that the range-midrange test is preferable to the Student $t$-test and the analogue of the Student $t$-test using the sample range (see [1] and [2]) whenever the sample size is sufficiently small. Use of the range-midrange test for the case of normality was proposed by E. S. Pearson in [3], where properties of the test were experimentally investigated for the normal and certain non-normal populations. In Part II several significance tests for the mean are developed which have a specified significance level for the case of a sample from a normal population but whose significance level is bounded near the specified value under very general conditions, one of which is that the observations are from continuous symmetrical populations. Some of these tests are range-midrange tests. Table 2 contains a summary of the tests and their properties ($x_i = i$th largest observation, $i = 1, \cdots, n$; conditions $(D)$ are given in section 7).

A method suggested by Wald for finding critical regions of almost constant size and various modifications are considered. Under reasonable conditions the $s$th step of this method gives a critical region of size $\alpha + R_s(\theta)$ where $\theta$ is the unknown value of the nuisance parameter, $R_s(\theta) = O(N^{-s/2})$ and $N$ is the sample size. The first step of this method gives the region which is obtained by assuming that an estimate $\hat \theta$ of the nuisance parameter is actually equal to $\theta$.

M. Loeve [3] has found conditions under which a sequence of events which may be interdependent in an arbitrary manner is strongly stable. In this note it is established that considerably weaker conditions imply the strong stability.

The fact that the mean and variance were sufficient statistics for a univariate normal distribution truncated at a fixed point was known to Fisher by 1931 [2]. Hotelling [3] has recently observed the corresponding fact for the truncated multivariate normal distribution. It is the aim of this note to point out that these are special cases of a general result, namely: If a family of distributions admits a set of sufficient statistics, then the family obtained by truncation to a fixed set, or by fixed selection, also admits the SAME set of sufficient statistics.

To determine the smallest sample size for which the minimum and the maximum of a sample are the $100 \beta %$ distribution-free tolerance limits at the probability level $\epsilon$, one has to solve the equation $N\beta^{N-1} - (N - 1)\beta^N = 1 - \epsilon$ given by S. S. Wilks [1]. A direct numerical solution of (1) by trial requires rather laborious tabulations. An approximate formula for the solution has been indicated by H. Scheffe and J. W. Tukey [2], however an analytic proof for this approximation does not seem to be available. The present note describes a graph which makes it possible to solve (1) with sufficient accuracy for all practically useful values of $\beta$ and $\epsilon$.