In an earlier paper, [1] an approach to the problem of systematic sampling was formulated, and the associated variance obtained. Several forms of the population were assumed. The efficiency of the systematic design as compared with the random and stratified random design was evaluated for these forms. It was remarked that as the size of sample increased the variance of a systematic design might also increase, contrary to the behavior of variances in the random sampling design. This possibility was verified in [2]. One approach to the study of systematic designs, given by Cochran [3] removed this difficulty to some extent by changing the problem to one of the expected variance, and supposing the elements of the population to be random variables. He showed that if the correlogram of these random variables is concave upwards, then the expected variance of the systematic design would be less, and often considerably less, than the variance of a stratified design. In the present paper the results of the earlier papers are extended to the systematic sampling of clusters of equal and unequal sizes. Some comments on systematic sampling in two dimensions are included. In section 2 we derive two theorems that have considerable applications in many parts of sampling. Although it has been common for people working in sampling theory to tell each other that these theorems ought to be true, yet no reference seems to exist. In section 3 we develop the implications of a remark [1, p. 13] that in designing sample surveys we should try to induce negative correlation between strata. In Theorem 3 we obtain sufficient conditions for the correlation to be negative. The lemma and Theorem 4 given in Section 4 enable us to extend the uses of Theorem 3 in practice. As an application of these results, we show that if a population has a concave upwards correlogram, and if strata are defined in an optimum fashion for the selection of one element at random from each stratum, then we can define a systematic type design that will be more efficient than independent random selection from each stratum. In sections 5 and 6 we obtain various results in the systematic sampling of clusters largely as applications of the more general theorems of the earlier sections. In general the results are of a nature similar to those of [1] and [3] in that the formulae show the conditions under which systematic sampling may be expected to be more efficient than random or stratified random sampling. We have not, however, applied these formulae to specified types of populations. From [1, 2 and 3] it is already apparent that this work will be useful and such studies should be more valuable when made in connection with important types of surveys or data than when made as illustrations in a general paper.

After consideration of the relative accuracies of systematic and stratified random sampling in one dimension the problem of estimation of linear sampling error is discussed. Methods of sampling an area are proposed, and expressions for the accuracies of these methods are derived. These expressions are compared for large samples, with special reference to correlation functions which appear to be theoretically and practically justified, and systematic sampling is found to be more accurate than stratified random sampling in many cases. Methods of estimating sampling errors are again considered, and examples given. The paper concludes with some remarks on the problem of trend in the population sampled.

The paper describes some results concerning the representation of a function by linear combinations of the successive differences of the Poisson distribution, not necessarily the partial sums of the type $B$ series of Charlier.

Asymptotic theorems on the difference between the (empirical) distribution function calculated from a sample and the true distribution function governing the sampling process are well known. Simple proofs of an elementary nature have been obtained for the basic theorems of Komogorov and Smirnov by Feller, but even these proofs conceal to some extent, in their emphasis on elementary methodology, the naturalness of the results (qualitatively at least), and their mutual relations. Feller suggested that the author publish his own approach (which had also been used by Kac), which does not have these disadvantages, although rather deep analysis would be necessary for its rigorous justification. The approach is therefore presented (at one critical point) as heuristic reasoning which leads to results in investigations of this kind, even though the easiest proofs may use entirely different methods. No calculations are required to obtain the qualitative results, that is the existence of limiting distributions for large samples of various measures of the discrepancy between empirical and true distribution functions. The numerical evaluation of these limiting distributions requires certain results concerning the Brownian movement stochastic process and its relation to other Gaussian processes which will be derived in the Appendix.

It is well known that the zero-order correlation between the predicted value of a variable and the observed value of the variable is the multiple correlation. It is also well known that the zero-order correlation between the residuals for two different variables, when the prediction is from a common set of variables, is the partial correlation. These considerations naturally lead to a systematic investigation of all the zero-order correlations involving the various variables associated with least squares theory. Such an investigation is the purpose of this paper. As a result of this study it appears that other zero-order correlations include the multiple alienation coefficient, the part correlation coefficient, and certain other coefficients which, as far as I am aware, have not been previously defined. The paper first examines the case of a single predicted variable and then continues with the case in which two or more variables are predicted simultaneously. The paper includes (1) a theoretical development of the different coefficients and the relations between them, (2) the expression of the formulas in determinantal form, (3) a matrix presentation of the material, and (4) an outline of the calculational techniques--with illustrations. It should be made clear at the start that this paper deals with populations (finite or infinite) and not with samples from those populations. The sampling distribution of each of the new correlation coefficients defined in this paper might well become the subject of a later investigation, but first we need to know what these correlation coefficients are.

The two inversion formulas considered here arise from study of G. A. Campbell's work on the Poisson summation, which is described more fully in the introduction and in the main consists of finding a function or mapping of a variable connected with the summation in terms of a normal (Gaussian) variable $g$. More generally, this last is a process often called "normalization of the variable" and associated with the names of E. A. Cornish and R. A. Fisher. The mapping is two-way and the main inversion formula determines co-efficients for one way from those for the other, both sets of coefficients being descriptive of their mappings. More precisely if $x$ is a given variable, $g$ a Gaussian variable, $y$ a parameter of the mapping, and the two mappings are $x = g + \sum^\infty_1 G_n(g) y^n/n!,$ $g = x + \sum^\infty_1 X_n(x)y^n/n!,$ the formula expresses $G_n(x)$ in terms of $X_i(x), i \leq n,$ and vice versa. The second formula is more particularly related to the Poisson summation and relates coefficients $p_n \equiv p_n(g)$ and $q_n \equiv q_n(g)$ in the pair of equations $a = c \sum^\infty_0 q_n c^{-\frac{1}{2}n}/n!$ $c = a \sum^\infty_0 p_n a^{-\frac{1}{2}n}/n!$ Both formulas, which are necessarily elaborate, are given concise expression by the use of the multi-variable polynomials of E. T. Bell.

In a previous paper [2] it was shown that it is sometimes profitable to select sampling units with probability proportionate to size of the unit. This note indicates a method of determining the probabilities of selection which minimize the variance of the sample estimate at a fixed cost. Some approximations that have practical applications are given.

By means of a general theorem, the space of the variables of classification is separated into population regions such that the probability of a correct classification is maximized. The theorem holds for any number of populations and variables but requires a knowledge of poppulation parameters and probabilities. A second theorem yields a large sample criterion for determining an optimum set of estimates for the unknown parameters. The two theorems combine to yield a large sample solution to the problem of how best to discriminate between two or more populations.

The fundamental identity is generalized to the case of independent random variables with non-identical distributions. The conditions for the validity of the differentiation of the identity are discussed. The results given in [1], [2], and [3] are obtained as special cases.