Some Concepts of Dependence

Problems involving dependent pairs of variables (X, Y) have been studied most intensively in the case of bivariate normal distributions and of 2 × 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 involving the general case center mainly around two problems: (i) tests of independence; (ii) definition and estimation of measures of association. In most treatments of these problems, there occurs implicitly a concept which is of importance also in other contexts (for example, the evaluation of the performance of certain multiple decision procedures), the concept of positive (or negative) dependence or association. Tests of independence, for example those based on rank correlation, Kendall’s Z-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. Similarly, 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 definitions of positive dependence, to investigate their consequences, explore the strength of each definition through a number of examples, and to give some statistical applications.