A "mixed model" is proposed in which the problem of the appropriate assumptions to make about the joint distribution of the random main effects and interactions is solved by letting this joint distribution follow from more basic and "natural" assumptions about the cell means. The expectations of the mean squares ordinarily calculated turn out, with suitable definition of the variance components, to have the same values as those usually found in more restrictive models, and some of the customary tests and confidence intervals are justified, but some aspects appear to be novel. For example, the over-all test found for the fixed main effects and the associated multiple-comparison method require Hotelling's $T$.

This is an attempt to present as simply as possible the best tools we know today for keeping computations simple when dealing with samples from general populations. Such computations seem inevitably to be made in terms of quantities related to moments. We develop here the formal structure and inter-relations of the two systems of multi-index quantities which seem today to be best adapted to statistical use. The occurrence of two systems is, at least in part, related to the appearance in statistical problems of both multiplication and addition of independent variables. Hence the existence of two systems, whose limiting cases are moments (about a fixed point) and cumulants (or semiinvariants). We present interconversion formulas, developing definitions and proving the pairing formulas without reference to any infinite populations, and sparing the use of combinatorial techniques as much as we are able. A few multiplication formulas are given, but for a more complete list the reader is referred to Wishart [10]. It is hoped that this paper can be read on its own, with some reference to applications of these techniques to elementary examples [6] and to the sampling properties of estimated variance components in the analysis of variance [7], [8], [9] as motivation. The author's best thanks go to N. R. Goodman for the checking of certain calculations.

A family of polynomials in the elements of a two-way array, or matrix, is introduced. This family is an extension, from sets to matrices, of the family of symmetric polynomials $k_1, k_2, k_{11}, k_3, k_{12}$, etc., defined by Tukey [6], christened "polykays" in [7], and which are a generalization of the family $k_1, k_2$, etc., defined by R. A. Fisher [1]. The polynomials of the present paper, called "bipolykays," are symmetric functions in the sense that they are invariant under permutation of rows and/or columns of the matrix. This paper defines the bipolykays, shows that they are inherited on the average, develops the formulas for use in random pairing, and provides tables for conversion and for multiplication. A description of applications (see [2], [3], and [4]) will be postponed until a later paper. These applications include (a) finding expressions for sampling moments of functions of the elements of a matrix which is a "bi-sample" from a larger matrix, (b) finding expressions for sampling moments of functions (such as estimates of variance components) associated with the analysis of variance of a two-way table with systematic interactions, and (c) finding unbiased estimators for the variances and covariances of estimated variance components in a two-way table without interactions.

Bipolykays were introduced in [3]. They form a family of symmetric (row-wise and column-wise) polynomial functions of the elements of a two-way array, with the property of being inherited on the average, and such that any similarly symmetric polynomial function of the same numbers can be written linearly in terms of the bipolykays. This paper will describe some applications of bipolykays to problems in the analysis of variance of two-way classifications, using the formulas and tables derived in [3]. A linear model which includes contributions from interaction as well as independently sampled cell contributions is given in Section 3, and applications are made to certain cases of this model. These applications include (a) finding unbiased estimators for the variance components in the case of no interaction as well as unbiased estimators for the variances of these estimators (Section 6), (b) finding expressions for means and variances of some of the functions of degrees 1 and 2 that are of interest in the problem of sampling from a matrix (Section 7), and (c) finding unbiased estimators for variance components in the general case, including expressions for the variances of these estimators in the case of infinite populations (Section 8).

Time series which are realizations of a vector-valued stochastic process of dimension two with a stationary disturbance are considered. Linear estimates of the regression coefficients of the time series are discussed, in particular the least-squares or classical estimate and the Markov estimate. The least-squares estimate is the estimate computed under the assumption that the components of the disturbance are orthogonal processes and orthogonal to each other. It is known that the Markov estimate is in general better than the least-squares estimate. The asymptotic behavior of the covariance matrices of the least-squares estimate and of the Markov estimate is described. Conditions under which the least-squares estimate is as good asymptotically as the Markov estimate are obtained, that is, conditions under which the least-squares estimate is efficient asymptotically in the class of linear unbiased estimates. The analogues of the results described for vector-valued time series of dimension greater than two can be seen to hold.

Certain results of information theory are applied to some problems of multivariate analysis, including the multivariate linear hypothesis and the hypothesis of homogeneity of covariance matrices. A discussion of certain related linear discriminant functions is also included. Some asymptotic distributions on the null hypothesis are derived. Related problems, still under investigation, are mentioned. The procedures are based on the principle of maximizing information. For the cases considered, the estimates of $I(1:2)$ and $J(1,2)$ turn out to be those obtained by replacing the parameters by unbiased estimates, appropriate to the hypotheses under consideration.

The authors continue the study (initiated in [1]) of the general queueing process (arbitrary distributions of service time and time between successive arrivals, many servers) for the case $(\rho < 1)$ where a limiting distribution exists. They discuss convergence with probability one of the mean waiting time, mean queue length, mean busy period, etc. Necessary and sufficient conditions for the finiteness of various moments are given. These results have consequences for the theory of random walk, some of which are pointed out. This paper is self-contained and may be read independently of [1]; the necessary results of [1] are quoted. No previous knowledge of the theory of queues is required for reading either [1] or the present paper.

In this paper definitions are given for three types of tolerance regions. For distribution-free tolerance regions, an analytic condition is derived for the characteristic function of the region. Examples of the application of the condition are considered. For $\beta$-expectation tolerance regions, a criterion for a good tolerance region is introduced, and it is shown that the problem of finding such a tolerance region can be reduced to that of finding a good test for an equivalent hypothesis-testing problem. Best tolerance regions are obtained for a number of single variate and multivariate problems involving normal distributions.

A method for constructing tolerance limits due to Fraser [8] is generalized by allowing that each step of the construction may depend not only on the blocks previously formed but also on all the known boundary observations and, moreover, on certain sets of indices. Furthermore, Tukey's [5] lexicographical ordering is replaced by a more general type of ordering.

A remarkable characterization of the normal law is that if $x$ and $y$ are two independent chance variables such that two linear functions, $ax + by (ab \neq 0)$ and $cx + dy (cd \neq 0)$, are distributed independently of each other, then both $x$ and $y$ are normally distributed. This theorem has been proved without any assumption about the existence of moments by Darmois [2], extending earlier results of Gnedenko [4] and Kac [5]. The question that naturally arises is how far the condition of stochastic independence is necessary, or, in other words, whether the above theorem can be generalised after relaxing the condition of stochastic independence of the linear functions of two independent chance variables. But it is evident that we can always construct two linear functions of non-normal mutually independent chance variables such that they are not independent in the probability sense. In the present paper we shall investigate the nature of the distribution law that may be obtained by imposing the mild restriction of the linearity of regression of one linear function on the other, which is, of course, weaker than the assumption of stochastic independence. We shall prove a general theorem from which a number of results will follow as special cases. But it should be noted that the statements regarding regression or conditional expectation require the assumption that the conditional distribution function exist, and in the following, this assumption will be tacitly made wherever needed.

It is the purpose of this paper to develop an expression for the probability of $x$ successes in $n$ trials, $P(n, x)$, where the probability of success on a single trial depends both on the number of the trial and on the number of previous successes. This result should prove useful in obtaining various probability functions. It will be noted that this work includes the case considered by Woodbury [3].

In many statistical problems, we obtain functions of both the random variable and the parameters involved, from whose asymptotic behavior we may deduce the asymptotic behavior of certain estimates. In many of these cases, it is sufficient to demonstrate uniform convergence with probability one of these functions. In this paper, a set of sufficient conditions for this is given, and we show how these results may be applied to some statistical problems.