POSITIVE AND NEGATIVE DEPENDENCE OF TWO RANDOM VARIABLES

Abstract

is an almost sure nondecreasing relation between them, and positively ?-dependent if their joint distribution is a mixture (with mixture coefficient k) of the distribution of two com letely positively dependent random variables and the distribution of two independent random variables with the same marginals. Similar definitions can be given for complete negative dependence and negative ?-dependence. The* paper discusses properties of such vari?tes and properties of the power of various tests for independence against such types of dependence. In another paper (Konijn, 1956), the author has investigated the asymptotic power of various tests for independence of two random variables against the alter native that their joint distribution has been obtained by a nontrivial linear transfor mation of independent random variables and has given some justification for consi dering such a type of alternatives. Among the many other types one could consider, it seems natural to examine one under which the joint distribution belongs to a family of bivariate distributions, all with the same marginals, which, as an extreme, contains one implying a monotone relation between the vari?tes. In fact, for any two given marginals, there is a unique joint distribution which implies a nondecreasing relation between the vari?tes, and a unique joint distribution which implies a nonincreasing relation. We shall use what is mathematically perhaps the simplest method of cons tructing such families of distributions, by linearly combining such an extreme and the distribution corresponding to independence with the same marginals. It is true that the bivariate distributions belonging to such families (other than those corresponding to the case of independence) all have a singular component, which seems rather unusual. Nevertheless, it may well be that a number of situa tions of practical interest can be represented approximately by a distribution of this family with the singular component replaced by a narrow band of probability mass about the corresponding curve in the plane of the two vari?tes. 2. Definition and chief properties of F+ and F_