The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. 1. Type $_0F_0$, exponential: (i) $\chi^2$, (ii) Wishart, (iii) latent roots of the covariance matrix. 2. Type $_1F_0$, binomial series: (i) variance ratio, $F$, (ii) latent roots with unequal population covariance matrices. 3. Type $_0F_1$, Bessel: (i) noncentral $\chi^2$, (ii) noncentral Wishart, (iii) noncentral means with known covariance. 4. Type $_1F_1$, confluent hypergeometric: (i) noncentral $F$, (ii) noncentral multivariate $F$, (iii) noncentral latent roots. 5. Type $_2F_1$, Gaussian hypergeometric: (i) multiple correlation coefficient, (ii) canonical correlation coefficients. The modifications required for the corresponding distributions derived from the complex normal distribution are outlined in Section 8, and the distributions are listed. The hypergeometric functions $_pF_q$ of matrix argument which occur in the multivariate distributions are defined in Section 4 by their expansions in zonal polynomials as defined in Section 5. Important properties of zonal polynomials and hypergeometric functions are quoted in Section 6. Formulae and methods for the calculation of zonal polynomials are given in Section 9 and the zonal polynomials up to degree 6 are given in the appendix. The distribution of quadratic forms is discussed in Section 10, orthogonal expansions of $_0F_0$ and $_1F_1$ in Laguerre polynomials in Section 11 and the asymptotic expansion of $_0F_0$ in Section 12. Section 13 has some formulae for moments.

Let $\{X_n, n = 0, \pm 1, \cdots\}$ be a real valued discrete parameter stationary stochastic process on a probability space $(\Omega, \mathscr{F}, P);$ for each $n = 1, 2, \cdots$, let $Z_n = \max (X_1, \cdots, X_n)$. We shall find general conditions under which the random variable $Z_n$ has a limiting distribution function (d.f.) as $n \rightarrow \infty$; that is, there exist sequences $\{a_n\}$ and $\{b_n\}, a_n > 0$, and a proper nondegenerate d.f. $\Phi(x)$ such that \begin{equation*}\tag{1.1}\lim_{n \rightarrow \infty} P\{Z_n \leqq a_nx + b_n\} = \Phi(x)\end{equation*} for each $x$ in the continuity set of $\Phi(x)$. The simplest type of stationary sequence $\{X_n\}$ is one in which the random variables are mutually independent with some common d.f. $F(x)$. In this case, $Z_n$ has the d.f. $F^n(x)$ and (1.1) becomes \begin{equation*}\tag{1.2}\lim_{n \rightarrow \infty} F^n(a_nx + b_n) = \Phi(x).\end{equation*} It is well known that in (1.2) $\Phi(x)$ is of one of exactly three types; necessary and sufficient conditions on $F$ for the validity of (1.2) are also known [9]. The three types are usually called extreme value d.f.'s [10]. Theorem 2.1 gives the limiting d.f. of $Z_n$ in a stationary sequence satisfying a certain condition on the upper tail of the conditional d.f. of $X_1$, given the "past" of sequence: the limiting d.f. is a simple mixture of extreme value d.f.'s of a single type. These are the same kind of d.f.'s found by us [3] to be the limiting d.f.'s of maxima in sequences of exchangeable random variables. The conditions of Theorem 2.1 are specialized to exchangeable and Markov sequences, and Theorem 2.2 extends the methods of Theorem 2.1 to general (not necessarily stationary) Markov sequences. It is shown that stationary Gaussian sequences, except for the trivial case of independent, identically distributed Gaussian random variables, do not obey the requirements of the hypothesis of Theorem 2.1: hence, Sections 3, 4, and 5 are devoted to a detailed study of the maximum in a stationary Gaussian sequence. Theorem 3.1 provides conditions on the rate of convergence of the covariance sequence to 0 which are sufficient for $Z_n$ to have the same extreme value limiting d.f. as in the case of independence, namely, $\exp (-e^{-x})$. The relation of these conditions to the spectral d.f. of the process is also discussed. A weaker condition on the covariance sequence ensures the "relative stability in probability" of $Z_n$ (Theorem 4.1). Theorem 5.1 describes the behavior of $Z_n$ when the spectrum has a discrete component with "not too many large jumps" and a "smooth" continuous component: when properly normalized, $Z_n$ converges in probability to a random variable representing the maximum of the process corresponding to the discrete spectral component. A special case was given by us in [2]. We now summarize some known results used in the sequel. The extreme value d.f.'s are continuous, so that (1.2) holds for all $x$; furthermore, this holds if and only if it holds for all $x$ satisfying $0 < \Phi(x) < 1.$ (1.2) implies that for all such $x$ $0 < F^n (a_nx + b_n) < 1,\quad\text{for all large} n,$ and \begin{equation*}\tag{1.3}\lim_{n \rightarrow \infty} F(a_nx + b_n) = 1.\end{equation*} Let $x_\infty$ be the supremum of all real numbers $x'$ for which $F(x') < 1$; then, for all $x$ satisfying $0 < \Phi(x) < 1$, we have \begin{equation*}\tag{1.4}\lim_{n \rightarrow \infty} a_nx + b_n = x_\infty.\end{equation*} From (1.3), and the asymptotic relation $-\log F \sim (1 - F), F \rightarrow 1$, we see that (1.2) holds if and only if \begin{equation*}\tag{1.5}\lim_{n \rightarrow \infty} n\lbrack 1 - F (a_nx + b_n)\rbrack = -\log \Phi(x).\end{equation*}

A problem of considerable importance in time series analysis is that of determining whether two Gaussian processes are equivalent, i.e., mutually absolutely continuous with respect to each other. If we are given $\{\Omega, \mathscr{a}, P_k\}, k = 0, 1$, where $\Omega$ is a set of real valued functions on some interval $\lbrack a, b\rbrack, \mathscr{a}$ is a Borel field of subsets of $\Omega$ and $P_k$ is a Gaussian probability measure on $\mathscr{a}$, then the problem is to determine the conditions under which $P_0$ is equivalent to $P_1$. Feldman (1958) has shown that a certain dichotomy exists in this problem in the following sense. If $\Omega$ is a linear space, then either $P_0$ and $P_1$ are equivalent or they are perpendicular, i.e., mutually singular. In addition, Feldman (1958) has shown, using some results of Segal (1958), that if $\Lambda$ is the linear span of $\Omega$ and the real constants, then $P_0$ and $P_1$ are equivalent iff the $P_0$-equivalence classes of $\Lambda$ are the same as the $P_1$-equivalence classes of $\Lambda$ and the identity correspondence between the $L_2(P_0)$ closure of $\Lambda$ and the $L_2(P_1)$ closure of $\Lambda$ is a bounded invertible operator $T$ such that $(T^\ast T)^{\frac{1}{2}} - I$ is a Hilbert-Schmidt operator. It is well known that a Gaussian process, and hence its probability measure, is uniquely determined by its covariance function, $R(s, t)$, assuming, of course, that the mean value functions is zero. It should, therefore, be possible to determine when $P_0$ and $P_1$ are equivalent in terms of the covariance functions of the two processes, $R_0(s, t), R_1(s, t)$. Indeed, it is desirable to have a theorem which states the conditions for equivalence of $P_0$ and $P_1$ in terms of the covariance functions, since in most applications the only information concerning the Gaussian processes consists of the covariance functions. In this sense, Feldman's (1958) theorem is in a form which is not too suitable for practical applications. This situation has been corrected by Feldman (1960) who, using his earlier results, has given the necessary and sufficient conditions for equivalence of $P_0$ and $P_1$ in terms of the covariance functions for a rather wide class of stationary Gaussian processes with zero mean value functions. However, Feldman (1960) has not computed the Radon-Nikodym derivative (RND) $dP_1/dP_0$ for pairs of equivalent Gaussian processes. The RND, or likelihood ratio, is of considerable importance in statistical inference for stochastic processes, as is well known. Recently, Parzen (1961), (1962) has described a new approach to time series analysis which is based on the notion of a reproducing kernel Hilbert space (RKHS). The theory of RKHS has been given by Aronszajn (1950). Using some results due to Aronszajn (1950) and Hajek (1958a), (1958b) Parzen (1962) has derived a necessary and sufficient condition for equivalence of pairs of Gaussian processes which have the same covariance function but have different mean value functions. In addition, he gives an explicit expression for the RND associated with pairs of equivalent Gaussian processes. The present work stems from a desire to derive the necessary and sufficient conditions for equivalence of pairs of stationary Gaussian processes with different covariance functions and zero mean value functions and to compute the RND for pairs of certain equivalent Gaussian processes. The method of proof is based on the RKHS approach used by Parzen. Thus, part of the results obtained will serve as an alternate derivation of Feldman's (1960) result. However, Feldman's proof involves a large number of complicated manipulations, while the present proof has the advantage of requiring only a few elementary manipulations. This simplification is achieved by employing convergence theorems of Aronszajn (1950), Hajek (1958a), (1958b) and Hormander (1963) and an expression for an inner product due to Parzen (1961).

The coefficient of coherence is defined for bivariate weakly stationary stochastic processes which have spectral distributions dominated by a fixed Lebesgue-Stieltjes measure. This quantity is shown to possess two of the important properties which make the ordinary correlation coefficient a desirable measure of linear regression for pairs of random variables. This provides a justification for the already common use of the coefficient of coherence as a measure of linear-regression for pairs of stationarily correlated, weakly stationary time series.

A population of particles is considered whose size $X_N(t)$ changes according to a branching stochastic process. The purpose of this paper is to find an approximate distribution of $X_N(t)$ when $t$ is fixed (not necessarily large) but the initial size of the population, $N$, is large. If $N$ is allowed to tend to infinity, and if the parameters of the process are made to change in a way analogous to the Poisson approximation of a binomial distribution, then it is shown that a limiting distribution of the process $X_N(t) - N$ exists as $N \rightarrow \infty$, and this limiting distribution is the distribution of a continuous process with independent increments. The relation between the parameters of the infinitely divisible distribution of the limiting process and the sequence of branching processes is exhibited.

The following queueing problem is considered. Customers arrive at a service facility at $r$ priority levels. At each priority level the input process is Poisson and these processes are mutually independent. The service times have an arbitrary distribution function which depends upon the priority level. A single server serves under a pre-emptive resume discipline. Results are obtained which characterize the transient and asymptotic distribution of the queue sizes and the waiting times. The analysis proceeds through reductions of the processes of interest to corresponding processes in a simple generalization of an M/G/1 queue.

In the present article, we shall propose a definition of the $r$th order interactions in a $m$-dimensional $d_1 \times d_2 \times \cdots \times d_m$ contingency table ($r = 0, 1, 2, \cdots, m - 1)$, and we shall present methods for testing the hypothesis that any specified subset of these interactions is equal to zero. In addition, we shall present simple methods for obtaining simultaneous confidence intervals for these interactions or for any specified subset of them. In the special case where the $m$-dimensional contingency table is a $2 \times 2 \times \cdots \times 2$ table (i.e., where $d_i = 2$ for $i = 1, 2, \cdots, m$), the $r$th order interactions defined herein are the same as Good's interactions [9], but the tests proposed by Good are different even in this case from those presented herein. When $d_i > 2$ for some values of $i$, Good's interactions are complex valued, whereas the interactions presented here are real valued. We shall show herein that the hypothesis $H_r$ that all $r$th order and higher-order complex interactions (defined by Good) are equal to zero is equivalent to the hypothesis $H^\ast_r$ that all $r$th order and higher-order real interactions (defined herein) are equal to zero, and that the test of $H_r$ within $H_s (r < s)$ presented by Good is asymptotically equivalent (under $H_r$) to the test of $H^\ast_r$ within $H^\ast_s$ presented herein. The tests presented herein are, in some cases, easier to apply than Good's tests. In addition, the methods presented herein are applicable to a wider range of problems in the sense that Good's methods can be used to test the null hypothesis that all $r$th order and higher-order interactions are equal to zero, whereas the methods presented herein can be used to test the more general null hypothesis that any specified subset of these interactions is equal to zero. The tests presented herein are generalizations of methods proposed earlier by Plackett [13] and Goodman [10] for testing the null hypothesis $H^\ast_2$ in a three-dimensional table. The test proposed by Good [9] is a generalization of the methods proposed earlier by Bartlett [5], Roy and Kastenbaum [14], and Darroch [8] for testing $H^\ast_2$ in the three-dimensional table. All of these earlier papers were concerned mainly with the testing of null hypotheses. In the present article, in addition to our treatment of hypothesis testing, we shall also present two different methods for obtaining confidence intervals for the $r$th order real interactions in the $m$-dimensional contingency table $(r = 0, 1, 2, \cdots, m - 1)$.

Suppose we are given $p$ objects to be weighed in $N$ weighings with a chemical balance having no bias. Let $X = (x_{ij})$ be the weighing design matrix, where $x_{ij} = + 1$ or $-1$ if the $j$th object is included in the $i$th weighing by being placed respectively in the left or right hand pan, and $x_{ij} = 0$ if the $j$th object is not weighed in the $i$th weighing. The weighing results may be represented by means of the following matrix equation, \begin{equation*}\tag{1.1} y = Xw + \epsilon,\end{equation*} where $y$ is the column vector of the results recorded in the $N$ weighings, $\epsilon$ is the column vector of the errors in these results and $w$ is the column vector of the true weights. Under the assumption that $\epsilon$ has mean $0_{N,1}$ and dispersion matrix $\sigma^2I_N$, where $0_{m\times n}$ is the $m \times n$ null matrix and $I_N$ is the $N$th order identity matrix, the normal equations estimating $w$ are given by the equation \begin{equation*}\tag{1.2}S\hat{w} = X'y,\end{equation*} where $S = X'X$ and $\hat{w}$ is the column vector of the estimated weights. The weighing designs problem was studied till now when $S$ is non singular (cf. Hotelling [1], Kishen [2], Mood [3], Raghavarao [4]). There is no recorded literature dealing with weighing designs whose $S$ is singular. Occasions may arise when the experimenter is faced with weighing designs whose $S$ is singular. We now define DEFINITION 1.1. A weighing design $X$ is said to be singular if the matrix $S$ is singular. The word "singular weighing design" in the above definition is somewhat misleading, but is used for the lack of a suitable word. Singular weighing designs may occur in the following cases. 1. Bad designing. As there are no best weighing designs tables, an experimenter who desires to use a particular order weighing design, has to construct one for himself before starting the weighing operations. In such circumstances bad designing may result in singular weighing designs. 2. Laboratory observations. Many scientists are of the opinion that they can achieve greater precision in their readings by repeating their experiments. The process of repeating weighing operations may also lead to singular weighing designs, when the number of independent linear weighing operations made is less than the number of objects. 3. Accidental. Though optimum or best weighing designs have been selected by the experimenter, accidentally some objects may fall down and break while taking the weighings. In that case, if the experimenter continues his weighing operations, putting $x_{ij} = 0$ for the broken objects, he may finally obtain a singular weighing design. For example, let an experimenter plan to weigh 5 objects in 5 weighings with the design $P_5$ of [4]. After two weighings, the first two objects fell down and broke. If he continues the weighing operations assuming $x_{ij} = 0 (i = 3, 4, 5; j = 1, 2)$, he gets a singular weighing design. This paper deals with two statistical questions. The first question concerns the estimability of individual weights of the objects and the second question considered is the taking of additional weighings so as to render the problem one of full rank and if possible so as to minimize the resulting generalized variance of the estimates, which is equivalent to Mood's efficiency definition. In Section 2, results on singular weighing designs together with some matrix lemmas are given, outlining the proofs wherever necessary. Necessary and sufficient conditions for the estimability of individual weights of the objects are given in Section 3 and the problem of taking additional weighings so as to obtain the estimates of all the objects is considered in Section 4.

The class of partially balanced incomplete block designs (PBIB) with more than two associate classes has not yet been explored to a great extent. In fact, only a few $m$-associate class PBIB's $(m > 2)$ are known explicitly. One way to obtain such designs is certainly by generalizing the well-known PBIB's with two associate classes. Among these particularly the Group Divisible PBIB's lend themselves rather obviously to a generalization in this direction. Roy [8] and Raghavarao [7] have generalized the Group Divisible design of Bose and Connor [1] to $m$-associate class designs. The idea of another type of Group Divisible PBIB's with three associate classes, given by Vartak [11], was extended to an $m$-associate class design by Hinkelmann and Kempthorne [5] which they called an Extended Group Divisible PBIB (EGD/$m$-PBIB). In this paper we shall investigate the EGD/$m$-PBIB in some detail. The definition and parameters of this design are given in Section 2. In Section 3 we shall prove the uniqueness of its association scheme. For a design given by its incidence matrix $\mathbf{N}$, the properties of the matrix $\mathbf{NN}'$ will be explored in Section 4. The eigenvalues of $\mathbf{NN}'$, its determinant and its Hasse-Minkowski invariants $c_p$ are obtained, and non-existence theorems are given. These theorems are illustrated by examples. An example of an existent EGD/$m$-PBIB plan is given.

Given any fraction of a factorial experiment in which the treatments either occur zero or one time, previous results were obtained on augmentation of the treatment design matrix, $X$, such that the product of the transpose and of the augmented matrix, $X_1 = \lbrack X'\vdots X'\lambda\rbrack'$, resulted in a diagonal matrix, and on a transformation of $X_1$ to another matrix $X_2 = FX_1$. In the present paper results are obtained on the evaluation of the variances of estimated effects under augmentation, on the existence and evaluation of $F$ and $\lambda$, on the determination of aliases of effects, and on the calculation of inverses for $\lbrack X'X\rbrack$ and for the information matrix $\lbrack X'_{22}X_{22}\rbrack$ for the deleted treatments.

In the present article, we shall present two different methods for obtaining simultaneous confidence intervals for judging contrasts among multinomial populations, and we shall compare the intervals obtained by these methods with the simultaneous confidence intervals obtained earlier by Gold [5] for all linear functions of multinomial probabilities. One of the methods presented herein is particularly suited to the situation where all contrasts among, say, $I$ multinomial populations may be of interest, where each population consists of, say, $J$ classes. The other method presented herein is suited to certain situations where a specific set of contrasts among these populations is of interest; e.g., where, for each of the $\frac{1}{2}I(I - 1)$ pairs of populations, the $J$ contrasts between the corresponding probabilities associated with the twopopulations in the pair are of interest. For judging all contrasts among the $I$ multinomial populations, the confidence intervals obtained with the first method presented herein have the desirable property that they are shorter than the corresponding intervals obtained with the method presented by Gold [5]. For judging the $\frac{1}{2}I(I - 1)J$ pair-wise contrasts between the multinomial populations, the confidence intervals obtained with the second method presented herein have the desirable property that they are shorter than the corresponding intervals obtained with the first method, for the usual probability levels. In the present paper we shall also solve a problem first mentioned by Gold [5] but left unsolved in the earlier article. Gold took note of the fact that, in the usual analysis of variance context, the simultaneous confidence intervals obtained by Scheffe [14] and by Turkey [15] for judging contrasts among the parameters have the desirable property that rejection of the homogeneity hypothesis by the usual $F$ or Studentized range test implies the existence of at least one relevant contrast for which the corresponding confidence interval does not cover zero (see, for example, [14], pp. 66-77). She also noted that a result analogous to the Scheffe-Tukey result had not yet been obtained for her simultaneous confidence intervals, and she stated that the difficulty seemed to be that the homogeneity test is based on a $\chi^2$ statistic with $(I - 1)(J - 1)$ degrees of freedom in the case where $I$ multinomial populations, each consisting of $J$ classes, are tested for homogeneity, whereas her confidence intervals were based upon the $\chi^2$ distribution with $I(J - 1)$ degrees of freedom. In the present article, one of the methods we shall present for obtaining simultaneous confidence intervals for the contrasts among the $I$ multinomial populations will be based upon the $\chi^2$ distribution with $(I - 1)(J - 1)$ degrees of freedom, and these intervals will have desirable properties somewhat analogous to those enjoyed in the analysis of variance by the Scheffe confidence intervals and by the Tukey confidence intervals. A modification of the usual test of the null hypothesis that the $I$ multinomial populations are homogeneous will be presented herein, and we shall show that this modified test will lead to rejection of the null hypothesis if and only if there is at least one contrast, of the kind presented herein, for which the relevant confidence interval does not cover zero.

In [5] and [6] the author proposed point estimates, tests and confidence procedures for the parameters in a linear model, which have the same asymptotic efficiency (relative to the corresponding classical methods) as the Wilcoxon test has relative to the $t$-test. Here, "asymptotic" refers to the case that the numbers of observations per cell tend to infinity; in practice, they should presumably be at least equal to four. In the present paper, we shall consider experiments with only one observation per cell. It then of course becomes necessary to impose some restrictions on the parameters of the model. We shall suppose that an experiment, concerned with various factors at several levels, is replicated at different levels of a nuisance factor (i.e. in different "blocks") and that this nuisance factor does not interact with the factors of interest. Nonparametric procedures for this situation were considered earlier by Friedman (1937) and in [3], where, however, only tests were proposed for the hypothesis of no effect of the factors of interest. In the present paper we shall be concerned primarily with estimating arbitrary contrasts in the factors of interest, and also with the problem of testing such contrasts. Let us assume for the observations $X_{i\alpha}(i = 1, \cdots, c; \alpha = 1, \cdots, N)$ the model \begin{equation*}\tag{1.1}X_{i\alpha} = \nu + \xi_i + \mu_\alpha + U_{i\alpha} (\sum \xi_i = \sum \nu_\alpha = 0)\end{equation*} where the $\xi$'s are the parameters of interest, the $\mu$'s are the effects of the nuisance factor (or factors), and the $U$'s are independently distributed according to a common continuous distribution $F$. Except for the assumption of equal sample sizes, this agrees with the model of [5] and [6] when the $\mu$'s are assumed to be zero. We shall be interested in inference methods concerning the $\xi$'s, and their asymptotic properties as $N \rightarrow \infty$.

It has been observed [1], [3] that in the one-sample case the distribution-free statistics in common usage are both SDF (strongly distribution-free) and of the form $\psi\lbrack F(X_1), \cdots, F(X_n)\rbrack$, where $X_1, \cdots, X_n$ is a random sample and $F$ is the hypothesized cpf (cumulative probability function); and it has been proved [1], [3] that the two properties above are equivalent in the one-sample case. In the multi-sample cases, one observes that except for the statistics of the Pitman conditional tests, which are not SDF, a major portion of distribution-free statistics (e.g. Kolmogorov-Smirnov, Cramer-von Mises, Wald-Wolfowitz, Mosteller-Turkey, Epstein-Rosenbaum, empty cell and the rank-sum types) in common usage have both the SDF and rank properties. Z. W. Birnbaum (in a personal communication in March, 1963) asks whether the SDF property implies the rank property in the two-sample case. An affirmative answer to this question and its converse would be of use both in analyzing and constructing multisample distribution-free statistics. In this paper, it is shown that in the multisample case, the rank property implies the SDF property; and that, except for zero-probability sets, the two properties are equivalent if the $k$-sample statistic $T$ satisfies Scheffe's [5] NB (null boundary) condition. The former result follows from the definitions of the rank and SDF properties. In proving the latter result one first shows that a completeness property of the class of strictly increasing continuous cpfs implies that each SDF, $k$-sample statistic $T$ is AI (almost invariant) in the appropriate sense; and, then, that the NB condition and AI property imply invariance and, hence, the rank property almost everywhere.

The exact and limiting distribution of quantiles in the univariate case is well known. Mood [3] investigated the joint distribution of medians in samples from a multivariate population, showing that their distribution is asymptotically multivariate normal. Recently Siddiqui [4] considered the joint distribution of two quantiles and an auxiliary statistic and showed that asymptotically their joint distribution is trivariate normal. Further, he showed the "distances" $X'_{i+l} - X'_i - X'_{i-h}$, ($l$ and $h$ fixed positive integers) between quantiles in the univariate case, when appropriately normalized are asymptotically independently distributed as Chi square r.v.'s with $2l$ and $2h$ d.f. respectively. In this paper the joint distribution of several quantiles from a bivariate population is obtained and it is shown that the distances between quantiles in the separate component populations are independent asymptotically.

In [6], Wald and Wolfowitz present a method for obtaining approximate two-sided tolerance intervals for a normal distribution. Some readers of [6] have been left with the impression that Wald and Wolfowitz proved that the confidence level attained by their approximation converges to the nominal confidence level with increasing sample size $N$, and that the difference is $O(1/N^2)$ (see, e.g., [3]). However, Wald and Wolfowitz did not consider that problem in their paper, nor does the problem seem to have been considered elsewhere in the literature. The principal result of Wald and Wolfowitz is given at the end of Section 1. In Section 2 it is shown that the confidence level attained by the Wald-Wolfowitz approximation does converge to the nominal confidence level, that the difference is $O(1/N)$, and that $1/N$ is the exact order of the rate of convergence except for the confidence level .5. This is a corollary of a more general result obtained by considering the case in which $s^2 \sim \sigma^2\chi^2_n/n$, and independently $\bar{x} \sim \text{normal} (\mu, \sigma^2/N)$, where $n$ is not necessarily equal to $N - 1$. It is found, perhaps surprisingly, that as $N \rightarrow \infty$, the confidence level attained by the obvious generalization of the Wald-Wolfowitz approximation converges to the nominal confidence level fastest when $n$ is fixed, the difference being $O(1/N^2)$ in this case. Furthermore, if $n$ increases "too rapidly" as $N \rightarrow \infty$, then the confidence level attained does not converge to the nominal confidence level. These results are consequences of the two theorems proved in this paper. Theorem 1 in Section 2 states that if $n/N^2 \rightarrow 0$, then the confidence level attained by the generalization of the Wald-Wolfowitz approximation converges to the nominal confidence level, that the difference is $O(n/N^2)$, and that, with certain unimportant exceptions, $n/N^2$ is the exact order of the rate of convergence. In Theorem 2, Section 5, it is shown that if $n/N^2 \rightarrow \infty$, then the confidence level attained by the generalization of the Wald-Wolfowitz approximation converges to a limit which does not depend on the nominal confidence level. A modification of the generalized Wald-Wolfowitz approximation which has the desired convergence property in this case is presented. Certain facts used in Section 2 are verified in Sections 3 and 4. In Section 6, an heuristic explanation of the observed asymptotic behavior is given, and the possibility of improving the Wald-Wolfowitz approximation is discussed briefly. The basic notation and method of determining error bounds in this paper are essentially the same as those employed by Wald and Wolfowitz in [6]. Sections 2, 3 and 4 of this paper are counterparts, respectively, of Sections 4, 8 and 9 of the Wald-Wolfowitz paper. Equation (19) in the present paper plays the role that the basic Equation (4.1) did in the Wald and Wolfowitz paper.

The analysis of a linear relation is considered, when there are replications and all the variables involved are subject to errors or fluctuations. A test based on the $F$ distribution is derived for testing the hypothesis that the unknown relation is a given linear relation. From this test a joint confidence region for the coefficients of the linear relation is derived. A confidence region for the linear relation is then defined as the set of all points which belong to hyperplanes not rejected by the test. The corresponding confidence coefficient is not known exactly, but it is known to be greater than a previously chosen $P$. In the non-degenerate case, the confidence region is a hyperboloid centered at the centroid of the given points, and it has the property that a hyperplane is not rejected by the test if and only if it is entirely contained in it. This confidence region estimation procedure is compatible with the maximum likelihood estimation of a linear relation, in the sense that the maximum likelihood hyperplane is contained in the confidence region for the linear relation, if this region is not empty.

Using Stein's [7] generalisation of a method of Lehmann and A. Birnbaum (A. Birnbaum [2]), we show that two of the tests commonly used in multivariate analysis of variance are admissible.

The present paper is an attempt to define sufficiency in simple terms in the theory of sampling. This definition is a suitable version of the existing notion of sufficiency as defined by Fisher, Halmos and Savage, Bahadur and others. The paper gives justification for the use of sufficient statistics in sampling theory. Applications to interpenetrating subsampling and two-stage sampling are given. In interpenetrating subsampling, it is proved that for estimating the population mean when the subsamples are drawn by simple random sampling without replacement, an estimator better than the usual overall average of subsample means is given by the average of distinct sample units. An improved estimator of the population variance is derived. In two-stage sampling where the first-stage units are drawn with unequal probabilities and second-stage units by simple random sampling (without replacement), two estimators of the population mean which are better than the estimator in current use are given.

The tail of the binomial distribution is defined as \begin{equation*}\tag{1}E(n, s, p) = \sum^n_{r=s} \binom{n}{r} p^r (1 - p)^{n-r}\end{equation*} with the restriction that $p$ be non-negative and less than (approximately) $s/n$. An estimate of $E(n, s, p)$ can be considered as the product of two separate estimates, an estimate of the size of the leading term in (1) and an estimate of the ratio of $E(n, s, p)$ to the leading term. This paper is concerned solely with the second estimate. That is, let \begin{equation*}\tag{2}R = E(n, s, p) \bigg/ \binom{n}{s} p^s(1 - p)^{n-s};\end{equation*} two estimates of $R$ are presented. In Section 2 an upper bound to $R$ is derived from a geometric series approach. The resulting bound is an improvement over previous results in that it is useful even when $p$ is near $s/n$, as opposed to simpler geometric bounds, such as that of Bahadur [1], which either blow up or become excessively large when $p$ is near $s/n$. The bound is given in Theorem 1, Equation (9). Section 3 discusses the error in using this bound. Section 4 gives a normal approximation to $R$, namely $R^\ast$ of Equation (18). Theorem 3 shows that the relative error of this estimate goes to zero as $s$ and $n - s$ go to infinity provided $0 \leqq p \leqq (s - 1)/(n - 1)$ and under a weak restriction on $s/n$ in the limit. The uniformity of this result with $p$ contrasts with the many normal approximations to $E(n, s, p)$, such as those of Bernstein, [2] Feller, [6], [7] Uspensky, [9] and Camp [3] in which the relative error becomes infinite as $p$ approaches zero. Section 5 presents a brief discussion of the behavior of $R^\ast$.

This paper is concerned with the theorem that the $X^2$ goodness of fit statistic for a multinomial distribution with $r$ cells and with $s$ parameters fitted by the method of maximum likelihood is distributed as $\chi^2$ with $r - s - 1$ degrees of freedom. Karl Pearson formulated and proved the theorem for the special case $s = 0$. The general theorem was formulated by Fisher [2]. The first attempt at a rigorous proof is due to Cramer [1]. A serious weakness of Cramer's proof is that, in effect, he assumes that the maximum likelihood estimator is consistent. (To be precise, he proves the theorem for the subclass of maximum likelihood estimators that are consistent. But how are we in practice to distinguish between an inconsistent maximum likelihood estimator and a consistent one?) Rao [3] has closed this gap in Cramer's proof by proving the consistency of maximum likelihood for any family of discrete distributions under very general conditions. In this paper the theorem is proved under more general conditions than the combined conditions of Rao and Cramer. Cramer assumes the existence of continuous second partial derivatives with respect to the "unknown" parameter while here only total differentiability at the "true" parameter values is postulated. There is a radical difference in the method of proof. While Cramer regards the maximum likelihood estimate as being the point where the derivative of the log-likelihood function is zero, here it is regarded as the point at which the likelihood function takes values arbitrarily near to its supremum. The method of proof consists essentially of showing that the goodness of fit statistic is a quadratic form in the observed proportions when the observed proportions are close to the expected proportions. The known asymptotic properties of the multinomial distribution are then used. The asymptotic efficiency of the maximum likelihood estimator is proved at the same time.

There have been several papers recently in the literature devoted to the subject of selecting a "best" population--see for example Gupta and Sobel (1962), Guttman (1961) and others. In these papers, the problem was analyzed from the sampling theory point of view. Except for some simple cases, this approach frequently leads to the problem of eliminating nuisance parameters. And, unless rather strong assumptions about certain of the parameters involved are made, the problem usually becomes intractable. In this paper, we consider certain best population problems adopting a Bayesian approach. As is well known, the modern Bayesian approach provides a logical framework for decision making under uncertainty--Savage (1954), Luce and Raiffa (1957), etc. Indeed, the adoption of this approach in the best population problem leads to intuitively satisfactory decision procedures in the presence of nuisance parameters. We consider a collection of $k$ populations $\prod_1, \cdots,\prod_i, \cdots, \prod_k$, where $\prod_i$ is distributed with probability density function $f(y \mid \theta_i)$ and $\theta_i$ may be vector-valued. Associated with the populations is a utility function $U(\theta_i)$. The best population is defined to be the one with the largest value among the $U(\theta_i)$. The Bayesian statistical decision procedure for choosing the best population is of course dictated by the principle of maximizing expected utility. Now it usually is the case that the experimenter's interest focuses on a specific criterion $h_i = g(\theta_i)$ where $g$ is known. For example, $g(\theta_i)$ might simply be the mean or the reciprocal of the variance of the $i$th population, $i = 1, \cdots, k$. It would then be natural for the experimenter to regard his utility function $U$ as a function of $h_i$, i.e., $U = U(h_i)$. In this paper, we will be mainly concerned with the case in which the criterion $g(\theta)$ is defined as $h = g(\theta) = \int^{a_2}_{a_1} f(y \mid \theta) dy$, for specified $a_1$ and $a_2$. The interval $(a_1, a_2)$ is sometimes referred to as a tolerance interval and the quantity $h$ is called the coverage of this interval. Considerations of tolerance intervals and their coverages frequently arise in engineering application [see for example Bowker and Lieberman (1959), and Ostle (1963)]. For instance, in the manufacture of a certain type of rivet, the product will only be of use if the diameter of the head of the rivet measures between certain specified limits. Or in the assembling of "stable" amplifiers, certain of the electronic tubes used in the amplifier must have transconductances that lie within specified limits. Hence, it is important for the manufacturer of these products to know what percentage of the items produced meet these required specifications. Suppose the manufacturer can choose among $k$ different processes to produce his product, and suppose further that his utility of the $i$th population $U(h_i) = h_i$ (that is, other considerations such as cost can be ignored). Naturally, he will wish to choose that process which gives the largest coverage to the specification interval. Of course, we are including the case that one process be capable of $k$ independent modifications. As an example, in the manufacture of precision items on a lathe, there may be a choice of different cutting tools, different coolants, and various speeds of rotation. Our interest in this type of problem is also motivated by the fact that the coverage criterion $h$ as defined above is usually a complicated function of the parameter(s) $\theta$. One might thus be led to suspect that it would in general be rather difficult to obtain optimal decision procedures for choosing the best population. In fact, one of the present authors--Guttman (1961)--has considered this problem within the sampling theory framework for the cases that $f$ takes the form of a normal density and that of an exponential density. It is evident from this earlier work that one would indeed encounter considerable mathematical difficulties unless certain specific assumptions about the scale parameters of the populations were made. We shall demonstrate, however, that no such difficulty exists in the Bayesian formulation and our analysis leads to results which seem intuitively satisfactory. In adopting a Bayesian approach, suppose we denote the prior distribution of the parameters of the $k$ populations under consideration as $p(\theta_1, \cdots, \theta_k)$. Then, for given independent samples from these populations, say $(\mathbf{y}_1, \cdots, \mathbf{y}_k)$, we can obtain the posterior distribution of these parameters. As we are interested in $(h_1, \cdots, h_k)$, which are themselves functions of the parameters, we may, in principle at least, determine their joint posterior distribution. This joint posterior distribution summarizes all the relevant information about $(h_1, \cdots, h_k)$. We assume throughout this paper that the population parameters $(\theta_1, \cdots, \theta_k)$ are locally independent a priori. This means that in the region in which the likelihood is appreciable, the joint distribution $p(\theta_1, \cdots, \theta_k)$ can be written approximately as the product $p_1(\theta_1) \cdots p_k(\theta_k)$. Such an assumption will be appropriate in situations where the prior distribution of the parameters is diffuse and gently changing over a wide region--see for example the discussion in Savage et al. (1962). It is clear that this independence assumption together with the assumption about the independence of samples implies that the $h$'s are locally independent a posteriori and hence can be analyzed individually. In Sections 2 and 3, we assume that the utility function $U(h_i)$ is $h_i$ for all $i$, and therefore analyze the problem by comparing the expected values of the posterior distribution of the coverages. The populations involved are assumed to be (1) normal and (2) exponential. The posterior distributions and the moments of the coverages for these populations are derived in Sections 4 and 5 where other types of utility functions are discussed. In Section 6 we briefly discuss some different types of best population problems.

The paper is mainly concerned with determining prior distributions on ignorance over parameter spaces, using invariance techniques similar to those of decision theory. Prior distributions are rarely determined exactly by such techniques and a number of less compelling methods for exact determination are given.

This paper extends some of the results obtained by Freedman [2]. In Section 1 a class of prior distributions on the space of all substochastic distributions on the positive integers is given, such that along almost all sample sequences the corresponding posterior distributions of the expectations of all bounded functions on the positive integers are asymptotically normal. Section 2 shows that most of Freedman's results carry over to the case of distributions on the closed unit interval.

The voting paradox, that it is possible among three candidates to have $A$ more popular than $B, B$ more popular than $C$, and $C$ more popular than $A$ (e.g., let $\frac{1}{3}$ of a population prefer $A$ to $B, B$ to $C$; another $\frac{1}{3}$ prefer $B$ to $C, C$ to $A$; and the remaining $\frac{1}{3}$ prefer $C$ to $A, A$ to $B$) naturally raises the question of how much more popular they can be, and what results can be obtained with more than three candidates. The question corresponds to the mathematical problem of choosing the joint distribution of $n$ real-valued random variables so as to maximize $\min\{P(X_1 > X_2), \cdots, P(X_{n-1} > X_n), P(X_n > X_1)\}$. The fact that all these probabilities can exceed $\frac{1}{2}$ is well known, (see [1]), but the question of max-min does not appear to have been considered. This note considers this problem (a) with unrestricted $X_1, \cdots, X_n$ and (b) with $X_1, \cdots, X_n$ restricted to be independent. In Case (a), it is very easy to show that the largest possible minimum is $(n - 1)/n$, easily achievable. In Case (b), which is more interesting, there is also an achievable largest minimum $b(n)$, which can be found by solving a degree $\lbrack\frac{1}{2}(n + 1)\rbrack$ equation, and has $b(n + 1) > b(n), \lim_{n\rightarrow\infty} b(n) = \frac{3}{4}$, and $b(3) = .61803, b(4) = \frac{2}{3}, \cdots, b(10) = .73205$.

A time series is the realization of a stochastic process. An estimate of the process average can be taken as the equally weighted average of observed values taken from the time series. Normally, the sequence of observations consists of observations taken at equal intervals of time. In certain applications, it is more appropriate to consider a sequence of observations spaced by intervals which are randomly and independently chosen from an exponential distribution. Let $\lbrack x_t, t \varepsilon T\rbrack$ be a real stochastic process that takes on values in a space $X, T$ being restricted to the real line. Let $\Omega$ be a space of points $\omega, F_T = \beta(x_t, t \varepsilon T)$ be the Borel field of $\omega$-sets generated by the class of sets of the form $\lbrack x_t(\omega) \varepsilon A\rbrack$ where $t \varepsilon T$ and $A$ is any Borel set, $P$ be the probability measure of $F_T$-sets, $\alpha$ be the Borel field of $X$-sets and for every real $t, x_t(\cdot)$ be a function from $\Omega$ to $X$ such that $\lbrack x_t(\omega) \varepsilon A\rbrack$ is an $F_T$-set for $A \varepsilon \alpha$. Assume that the process satisfies, for all $t \varepsilon T$, the conditions \begin{align*}\tag{1.1}\quad E\{x_t\} &= \bar x \quad \text{exists and is independent of} \quad t \\ \tag{1.2}E\{(x_t - \bar x^2\} &= \operatorname{var} x \quad \text{exists and is independent of} \quad t \\ \tag{1.3} E\{x_t - \bar x) (x_{t + \tau} - \bar x)\} &= r \quad (\tau) \operatorname{var} x \quad \text{depends only on} \quad \tau \quad \text{and} \\ r \quad (\tau) \text{is harmonizable}. \end{align*} A stochastic process satisfying Conditions (1.1) to (1.3) is often referred to as being stationary in the wide sense [2]. \begin{equation*}\tag{1.4}\lim_{\tau\rightarrow\infty} \tau r(\tau) \rightarrow 0.\end{equation*} Let $\{\xi\}$ be a measurable space of mutually independent random variables $\xi_i$ identically distributed with probability $p$ of having value 1 and $(1 - p)$ of having value 0, all independent of the $x_t$-process. The sample space $\Omega$ contains all possible realizations of the stochastic process. Given a scheme for observing the values of $x_t$ in time and some form of estimate of process mean from the observed values, expected values of such an estimate and of its square can be determined. A measure of the efficiency of the sampling scheme may be defined as the ratio of the variance of the estimator to that which would be obtained if the observations taken were independent of each other. The limit in probability of this measure of sampling efficiency for observations spaced by exponentially distributed mutually independent intervals is obtained simply in terms of the mean observation rate and the spectral density component at zero frequency. The limit theorem considered is a weak version. By augmenting Condition (1.4) with relations between sample size $N$, mean observation rate $\lambda$, length of time series $T$ and sufficiently rapid decay of $r(\tau)$ with $\tau$, sharper probability statements can be made.

In an interesting recent article [4], J. W. Moon has given a list of properties of the graph $L(B_{mn})$ (to be defined more precisely below) and investigated the question of whether these properties characterize the graph. In case $m = n$, this question had been settled by Shrikhande [5] (see also [1], [2] and [3]), who proved that the answer was yes unless $m = n = 4$, when there is exactly one exception. In case $m > n$, Moon shows that the answer is yes unless $(m, n) = (5, 4)$ or (4, 3), which cases were left unsettled in [4]. The purpose of this note is to show that, in those cases as well, the answer is yes, thus completing Moon's discussion. Now to define our problem more exactly. The line graph of the complete bipartite graph on sets with $m$ and $n$ vertices, denoted by $L(B_{mn})$, is the graph with $mn$ vertices given by all ordered pairs $(i, j), 1 \leqq i \leqq m, 1 \leqq j \leqq n$. Two vertices $(i, j)$ and $(i', j')$ are joined by an edge if $i = i'$ or $j = j'$, but not both. The graph $L(B_{mn})$ has the following properties: (1.1) It has $mn$ vertices. (1.2) Each vertex has valence $m + n - 2$. (1.3) If two vertices are not adjacent, there are exactly two vertices adjacent to each. (1.4) Of the $\frac{1}{2} mn(m + n - 2)$ pairs of adjacent vertices, exactly $n\binom{m}{2}$ pairs are each adjacent to exactly $m - 2$ vertices, the remaining $m\binom{n}{2}$ pairs are each adjacent to exactly $n - 2$ vertices. We now assume $m > n$, and let $G_{mn}$ be any graph satisfying (1.1)-(1.4). Our object is to prove that when $(m, n) = (5, 4)$ or (4, 3), $G_{mn} = L(B_{mn})$. Moon has established $G_{mn} = L(B_{mn})$ in all other cases.

The problem of two-sample rank order tests is examined from the point of view of I. R. Savage (1956, 1957, 1959, 1962). Under suitable restriction of the class of alternatives some rank orderings are always more probable than (dominated by) others. Hence, if the rejection region of a test contains the ordering $c$ it must also contain all orderings dominated by $c$ if the test is to be admissible. Thus, by counting the number of orderings dominated by $c$ we arrive at a lower bound for the size of an admissible test which rejects when $c$ is observed. Similar reasoning leads to an upper bound. The counting is achieved by putting all orderings in one-one correspondence with paths on a grid; all paths lying below or along the observed path correspond to the orderings dominated by the observed ordering. An expression for this number of paths is obtained. This expression is used to compute significance bounds for a pair of illustrative examples. Finally, the main result for the two-sample problem is extended to obtain upper and lower bounds for a one-sample problem.

The normal equations in the analysis of variance with suitable side conditions give a unique set of estimates, $\tilde\beta_i$, of the parameters $\beta_i$. These estimates are unique linear forms in the actual observations. In the solutions to the normal equations they appear as linear forms in the treatment and block totals; these totals are not independent and so the forms in them are not unique. Thus the normal equations, while giving a unique solution vector, admit an infinite number of pseudo-inverses of the matrix $X' X$. In this paper the relationship between the two most common pseudo-inverses is discussed.