Some Concepts of Dependence

Abstract

Problems involving dependent pairs of varia- bles (X, Y) have been studied most intensively in the case of bivariate normal distributions and of 2 X 2 tables. This is due primarily to the importance of these cases but perhaps partly also to the fact that they exhibit only a particularly simple form of dependence. (See Examples 9(i) and 10 in Section 7.) Studies in- volving the general case center mainly around two problems: (i) tests of inde- pendence; (ii) definition and estimation of measures of association. In most treatments of these problems, there occurs implicitly a concept which is of im- portance also in other contexts (for example, the evaluation of the performance of certain multiple decision procedures), the concept of positive (or negative) de- pendence or association. Tests of independence, for example those based on rank correlation, Kendall's t-statistic, or normal scores, are usually not omnibus tests (for a discussion of such tests see [4], [15] and [17], but designed to detect rather specific types of alternatives, namely those for which large values of Y tend to be associated with large values of X and small values of Y with small values of X (positive dependence) or the opposite case of negative dependence in which large values of one variable tend to be associated with small values of the other. Simi- larly, measures of association are typically designed to measure the degree of this kind of association. The purpose of the present paper is to give three successively stronger defi- nitions of positive dependence, to investigate their consequences, explore the strength of each definition through a number of examples, and to give some statistical applications. 2.