Positive Dependence Orders

Abstract

Notions of positive dependence of two random variables X1 and X2 have been introduced in the literature in an effort to mathematically describe the property that “large (respectively, small) values of X1 tend to go together with large (respectively, small) values of X2.” Many of the notions of positive dependence are defined by means of some comparison of the joint distribution of X1 and X2 with their distribution under the theoretical assumption that X1 and X2 are independent. Often such a comparison can be extended to general pairs of bivariate distributions with given marginals. This fact led researchers to introduce various notions of positive dependence orders. These orders are designed to compare the strength of the positive dependence of the two underlying bivariate distributions. In this chapter we describe some such notions.