A class of statistics, large enough to comprise those used in all the known distribution-free tests of fit for continuous distribution functions, is characterized by a structure called "structure ($d$)." A number of statistics of this class may be constructed and used for tests of fit. To make a reasonable choice among all these statistics, it appears desirable to introduce in the space of continuous distribution functions a distance which would reflect the type of discrepancy the proposed test is intended to detect. By studying the power of various statistics with regard to this distance one may then be able to choose those with optimal properties.

A new mathematical model is proposed for the action of counters such as the Geiger-Mueller or the scintillation counters. It is assumed that after each registration the counter is inoperative for a time interval of random length. The distribution of lengths of the inoperative periods is so defined that the Type I and Type II models familiar in the literature on counters are special cases. More important, it also allows an action that is a compromise between those two types. Assuming that the sequence being counted is a Poisson process with mean rate of occurrence $mT, m > 0$, in an arbitrary interval of length $T$, the process generated by the counter is discussed and rules are established for obtaining confidence intervals for the parameter $m$ from various types of counting experiments.

Simple nonparametric classes of alternatives are defined for various nonparametric hypotheses. The power of a number of such tests against these alternatives is obtained and illustrated with some numerical results. Optimum rank tests against certain types of alternatives are derived, and optimum properties of Wilcoxon's one- and two-sample tests and of the rank correlation test for independence are proved.

Nonparametric tolerance regions can be constructed from statistically equivalent blocks using published graphs by Murphy [8]. In this paper the procedure for obtaining the statistically equivalent blocks is generalized. The $n$ `cuts' used to form the $n + 1$ blocks need not cut off one block at a time, but at each stage may cut off a group of blocks, the group to be further divided at a later stage by a different type of cut in general. An example is given which indicates possible applications. The results are also interpreted for discontinuous distributions by indicating the necessary modifications to the corresponding theorem in [7].

Let $P$ be a finite population which may have some subsets of known size but whose total size $N$ is unknown. We shall consider the problem of point and interval estimation, tests of hypotheses, and fiducial distributions of $N$ for some sampling-tagging procedures. The problem of estimating the number of classes in a population [1], [2], when it is known that the same number of elements is contained in each class, may be considered within the general problem discussed.

The purpose of this paper is to prove certain results concerning complete classes of strategies, some of which were announced in an abstract in Bull. Am. Math. Soc., Vol. 57 (1951), p. 372.

Over the past fifteen or so years, a large number of classes of experimental designs have been evolved by Yates, Bose and Nair and others (see [1] for a systematic account). The aim in all cases was to evolve patterns of observation which could utilize natural groupings in the experimental material, such as for instance litters of mice or small numbers of plots perhaps contiguous to each other. By arranging the treatments to be compared in specific ways which utilize the natural grouping, it is possible to enable treatment contrasts to be estimated by comparisons of observations on experimental units, which we shall call plots, within the natural groups. This enables the comparisons to be made usually with considerably greater accuracy than would obtain if the experimenter were forced to randomize the positions of the treatments without respect to these groupings. In experimental work in some branches of biology, natural groups of size two are of fairly frequent occurrence, for example twins, or halves of plants, or halves of leaves. The development of experimental designs is not complete in this particular respect. The designs which have been developed for blocks of two plots or experimental units are as follows: (1) symmetrical pairs (Yates, [2]) which require $(t - 1)$ replicates if there are $t$ treatments. (2) quasifactorial designs if the number of treatments is a power of 2 (see [1] in this respect). It appears therefore that development of a class of designs using blocks of two plots is desirable, and this is the purpose of the present paper.

Among the methods of obtaining satisfactory parameter estimates are maximum likelihood, minimum chi-square, minimum "reduced" chi-square, etc. This paper presents a generalization of the minimum chi-square method which yields regular best asymptotically normal (RBAN) estimates and which is often very simple to apply. It is shown that the least squares expressions associated with the logit and probit transformations are a type which lead to RBAN estimates.

Let $X_1, X_2, \cdots, X_n$ be independent with a common distribution function $F(x)$ which has a finite mean, and let $Z_{n1} \leqq Z_{n2} \leqq \cdots \leqq Z_{nn}$ be the ordered values $X_1, \cdots, X_n$. The distribution of the $n$ values $EZ_{n1}, \cdots, EZ_{nn}$ on the real line is studied for large $n$. In particular, it is shown that as $n \rightarrow \infty$, the corresponding distribution function converges to $F(x)$ and any moment of that distribution converges to the corresponding moment of $F(x)$ if the latter exists. The distribution of the values $Ef(Z_{nm})$ for certain functions $f(x)$ is also considered.

The main result obtained is the following: If a population has monotone decreasing correlogram, then centered systematic sampling is more efficient than random start systematic sampling. It is also shown that if a population is monotonic, then centered systematic sampling is more efficient than random start systematic sampling, but here it is easy to cite cases in which stratified random sampling is more efficient than either. Thus, centered systematic sampling is more efficient than random start systematic sampling, in the conditions (namely, concave upwards and decreasing correlogram) in which Cochran [1] proved that random start systematic sampling is more efficient than stratified random sampling.

Theorem 1 provides us with a result from which we can derive the modified Schwarz inequality (1.2) or, more generally, (3.8). The formulas hold when $x(t)$ is any nondecreasing function belonging to a certain wide class, and $\bar{\varphi}(t)$ is the right-hand derivative of the "greatest convex minorant" of $\Phi(t)$. The necessary and sufficient conditions for equality to hold are also given. Applications to distribution problems in statistics are discussed in Section 4.

In the past, the principles which have been applied most often in the selection of an estimate are the principles of maximum likelihood and of minimum variance unbiased estimation. Recent statistical literature (e.g., [1]) has pointed out the fact that, while these principles are intuitively appealing, neither of them can be justified very well in a systematic development of statistics. Abraham Wald [2] has indicated a more systematic approach to the problem. One of Wald's ideas may be paraphrased as follows. Consider a random variable $X$ whose distribution depends on an unknown real parameter $\theta$. If the value $x$ of $X$ is observed one makes an estimate, say $f(x)$, and thereby incurs a loss of $W\lbrack\theta,f(x)\rbrack$. The risk associated with the estimate $f$ is defined to be the expected loss $R_f(\theta) = E\{W\lbrack\theta, f(X)\rbrack\mid\theta\}$. In choosing between two estimators $f_1$ and $f_2$, it seems clear that one would prefer $f_1$ to $f_2$ if $R_{f_1}(\theta) \leqq R_{f_2}(\theta)$ for all values of $\theta$, and $R_{f_1}(\theta) < R_{f_2}(\theta)$ for at least one value of $\theta$. We shall consider only the case where the loss as a function of $\theta$ and the estimate $f(x)$ is of the special form \begin{equation*}\tag{1}W\lbrack\theta,f(x)\rbrack = \lambda(\theta)(f - \theta)^2;\lambda(\theta) > 0.\end{equation*} Reasons for considering this form of $W\lbrack\theta,f(x)\rbrack$ have been given in [3]. Suppose that we know of an unbiased estimate $f$ whose variance is $K\theta^2$, where $K$ is known. Then, as we shall see, the risk of $f$ is greater than the risk of $f/(K + 1)$. Hence $f/(K + 1)$ is to be preferred to $f$ as an estimator of $\theta$. This result holds for any function $\lambda(\theta) > 0$. Although $f/(K + 1)$ is generally not unbiased in the usual sense, it is unbiased in a certain sense (cf. [4]). It is seen that a special case of this result is related to the problem of the estimation of the scale parameter of a population whose form is not given but for which the ratio of the first and second moments is known (cf. [5]). Some special cases and applications are discussed in detail.

The joint sampling distribution of $\bar{x}$ and $S$ is derived in integral form for probability density functions of doubly infinite range. This derivation is effected through the use of a transformation which transforms the sample probability element $f(x_1)f(x_2) \cdots f(x_n) dx_1 dx_2 \cdots dx_n$ into the element $f(x_1)f(x_2) \cdots f(x_{n - 2})f((n\bar{x} - \sum^{n - 2}_1 x_i \pm \Omega_1)/2)f((n\bar{x} - \sum^{n - 2}_1 x_i \mp \Omega_1)/2) |J| \\ \cdot dx_1 dx_2 \cdots dx_{n - 2}d\bar{x} dS,$ where $\bar{x} = (1/n) \sum^n_1 x_i, S^2 = (1/n) \sum^n_1(x_i - \bar{x})^2$, and $J$ is the Jacobian of the transformation. Bounds on $x_{n - r}, r = 2, 3, \cdots, n - 1$, are established in terms of $\bar{x}, S$, and $x_{n - r -j}, j = 1, 2, \cdots, n - r - 1$. The probability element $f(x)_1f(x_2) \cdots f(x_{n - 2})f((n\bar{x} - \sum^{n - 2}_1 x_i \pm \Omega_1)/2)f((n\bar{x}\\ - \sum^{n - 2}_1 x_i \mp \Omega_1)/2)|J|dx_1 dx_2 \cdots dx_{n - 2}d\bar{x} dS$ must then be integrated with respect to $x_{n - r}, r = 2, 3, \cdots, n - 1$, between these limits to obtain $F(\bar{x},S)d\bar{x}dS$, the joint probability element of $\bar{x}$ and $S$. These limits of integration of $x_{n - r}, r = 2, 3, \cdots, n - 1$ enable one to express $F(\bar{x}, S)$ in terms of quadratures when $f(x)$ is any probability density function of doubly infinite range. To illustrate the method, $F(\bar{x}, S)$ is obtained when $f(x)$ is the normal probability density function.

With the balanced incomplete block designs [1] $r$ replicates of each of $v$ treatments are arranged in $b$ `blocks', and the number of `plots', $k$, in each block is smaller than $v$. In order to eliminate systematic block differences from the comparison of treatment means and to obtain treatment comparisons of equal precision the well known conditions of balance specify (a) that no treatment should occur more than once in any block, (b) that the number of blocks in which any two particular treatments are both applied should be a constant number of blocks ($\lambda$ blocks) for all possible treatment pairs. When these designs are used in this form, the position of treatments within each block is not specified and can normally be regarded as unimportant. Accordingly the treatments in each block are randomized. Situations, however, arise in which the `plots' occupy certain characteristic positions in each block. Thus, if in a field experiment the blocks are vertical columns the plots would fall into $k$ horizontal rows which may also have systematic effects on the yields. In this case it will often be advantageous to balance the design with regard to rows (as well as with regard to columns) in a manner similar to the Latin square. Such an arrangement was first developed by Youden [2] who used the particular incomplete block designs with $b = v$ and in these rearranged the treatments in each block in such a way that every treatment occurred precisely once in each row. More recently one of us (S.S.S.) has carried out similar rearrangements for the other incomplete block designs with $b > v, v \leqq 10, r \leqq 10$ (ie., for those tabulated in standard tables and book), and has used a definition of balance resulting in at most two different precisions for treatment comparisons. In this note we show that (i) balancing with regard to rows resulting in an equal precision of all treatment comparisons is possible if $b = mv$ ($m$ integral), (ii) in all incomplete block designs with $r = mk \pm 1$ a row balance is possible resulting in treatment comparisons of two different precisions. One of us (W. B. T.) has prepared complete tables of double balanced designs suitable for practical use which it is hoped to publish together with the analysis of variance procedure with recovery of interblock information.*

A lower bound is derived for the expected number of observations required by an arbitrary sequential test which satisfies conventional conditions regarding the probabilities of erroneous decisions.