The distribution of the statistic $X$ which is the number of edges in the intersection graph $G_1 \cap G_2(V, E_1 \cap E_2)$ of $G_1(V, E_1)$ and $G_2(V, E_2)$ is investigated through its moments. An expression is obtained for the $r$th central moment and the moment ratios of $X$ are, under a set of sufficient conditions, shown to approximate to those of a normal variable with the standardised variable. $Z = \{X - \epsilon(X)\}/(\operatorname{var} (X))^{\frac{1}{2}}$ having an asymptotically unit normal distribution.

Define \begin{equation*}\tag{1.1}P = U/V \equiv \sum^m_{i=1}\lambda_ix_i^2/\sum^r_{i=1}x_i^2,\quad 1 \leqq m \leqq r,\end{equation*} where the $x_i$'s are independent and identically distributed as $N(0, 1)$, and $0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_m$. We will give various representations for the distribution of $P$, and show how a special case of this distribution is useful for testing for correlation in a time series. We will also consider independence of the errors in the normal regression problem. Let $n$ be the number of observations, and $k$ the number of coefficient parameters in a univariate linear regression model. Take $r = n - k$ and $m = r - 1$. Suppose the $x_i$'s are the linear functions of the observations of the dependent variable obtained by a "Theil transformation" (see Theil [15]). It will be shown in Section 2 that under these conditions, $P$ is an appropriate test statistic for independence. In Section 3, two different characteristic function representations are developed for $P$; one involves an infinite series of complex valued gamma functions, while the other involves a doubly infinite series of real valued gamma functions. Section 4 discusses the fact that for $r = m + 1, P$ is distributed as a linear combination of correlated beta variates. An appropriate beta distribution approximation is given for comparison. A numerical tabulation of the distribution of $P$ for the case of $r = m + 1$ is in preparation. The cdf. of $P$ is found by numerically inverting the characteristic function of a related linear combination of central chi-square variates.

When a hypothesis is tested by a significance test and is not rejected, it is generally agreed that all hypotheses implied by that hypothesis (its "components") must also be considered as non-rejected. However, when the hypothesis is rejected the question remains as to which components may also be rejected. Various writers have given attention to this question and have proposed a variety of multiple comparisons methods based either on tests of each one of the components or on simultaneous confidence bounds on parametric functions related to the various hypotheses. An approach to such methods, apparently originally due to Tukey [27], is to test each component hypothesis by comparing its statistic with the $\alpha$ level critical value of the statistic for the overall hypothesis. This is called a Simultaneous Test Procedure (STP for short) as all hypotheses may be tested simultaneously and without reference to one another. An STP involves no stepwise testing of the kind employed by some other methods of multiple comparisons for means, in which subsets are tested for equality only if they are contained in sets which have already been found significant. (See [3], [4], [10], [18]). A general formalization of STP's is attempted in this paper. Section 2 introduces the requisite concepts of families of hypotheses and the implication relations between them, as well as the monotonicity relations between the related statistics. Section 3 defines STP's and shows conditions for coherence and consonance of their decisions, these properties being that hypothesis implication relations are preserved in the decisions of the STP. Section 4 discusses comparison of various STP's for the same hypotheses and shows the advantages of the union-intersection type of statistics and of reducing the family of hypotheses tested as much as possible. Section 5 translates all these results to simultaneous confidence statements after introducing the definitions necessary to allow such translation. The analogy between simultaneous test and confidence methods is of special importance as it brings a wide spectrum of methods within this framework, most of which was originally formulated in confidence region terms. This covers the original work of Tukey [27] and Scheffe [25] and continues with that of Roy and his associates [21] and most recently Krishnaiah [12], [13]. A general discussion of this confidence approach has been given by Aitchison [1] since the first draft of the present paper. In view of the close analogies pointed out in Section 5, it is not surprising that Aitchison arrives at the requirement of a constant critical value for all his tests, exactly as in an STP. In fact his approach and the present one are complementary. The formal theory is illustrated with ANOVA examples to clarify the concepts. No essentially new techniques are presented in this paper though this approach has been used elsewhere by the author to derive a number of practically useful procedures [6], [7], [8].

Interpolation and extrapolation of stationary stochastic processes has been extensively studied by Kolmogorov, Wiener and Masani, Krein and many others. Rozanov [3] has formulated many of their results and some of his own in a very neat form. In this paper some of the basic concepts and theorems related to interpolation are investigated in the more general setting of homogeneous random fields on locally compact abelian groups. In the stationary case, the regularity and singularity of the process is determined by its behavior on the class of intervals $(-\infty, t\rbrack$. Here, since the group is not necessarily ordered, this class is replaced by an arbitrary family, $I$, of non-empty Borel sets of the group. Regularity and singularity are then defined in terms of the behavior of the field on the sets of $I$. Theorems 4.1 and 5.1 generalize Kolmogorov's minimality problem [1] and an interpolation problem studied by Yaglom [6] to groups. Theorem 4.1 is also seen to include the result of Wang Shou-Jen on interolation in $R_K$ [5]. The family $I_\infty$, introduced in Section 5, provides a natural generalization of the intervals $(-\infty, t\rbrack$ for certain processes.

In testing whether a treatment has an effect or not, the experimenter is often obliged to use the same subjects for control and treated groups. In such a case it is generally unrealistic to assume independence and one is led to tests of bivariate symmetry. The object of this paper is to show that bivariate symmetry is not equivalent to univariate symmetry; that there exists a feasible procedure different from the likelihood ratio test in the normal case; that there is no unique "natural" concept of rank; that all distribution-free (DF) procedures are based on permutations; and that optimal DF procedures for a simple alternative are based on permutations of the likelihood function.

The problem of approximating a point $\theta$ of minimum of a function $f \varepsilon \mathscr{C}$ (see 2.1) is considered. An approximation procedure of the type described in Fabian (1967) using the design described in Fabian (1968), but with the size of design increasing, achieves the speed \begin{equation*}\tag{1}E|X_n - \theta|^2 = o(t^{-1}_n \log ^3 t_n);\end{equation*} here $X_n$ is the $n$th approximation and $t_n$ the number of observations necessary to construct $X_1, X_2, \cdots, X_n$.

This paper deals with the structure of sets $\Omega$ of distributions for which a particular test is the most powerful for testing a simple hypothesis $H:f = f_0 \operatorname{vs.} K:f \varepsilon\Omega$, that is, with the domain of optimality of a test. The context is restricted to these $\Omega$ consisting of probabilities having continuous positive densities, and to one-sample tests. The important concept is that of a family of tests, one for each significance level. This concept allows us to use the full power of the Neyman-Pearson Lemma. The main results are: (1) The domain of optimality of a test family $\Phi$ is essentially a multiplicatively-convex (convex in the logarithms) cone; hence there are distributions both "near to" and "far from" the null distribution for which $\Phi$ is optimal. (Theorems 1, 2, and 3). (2) If $\Phi$ is uniformly most powerful for testing $H:f = f_0 \operatorname{vs.} K:f \varepsilon\Omega$ with $n \geqq 2$ then the class of distributions has a monotone likelihood ratio. (Theorem 4).