Almost opposite regression dependence in bivariate distributions

Abstract

Let X,Y be two continuous random variables. Investigating the regression dependence of Y on X, respectively, of X on Y, we show that the two of them can have almost opposite behavior. Indeed, given any e > 0, we construct a bivariate random vector (X,Y) such that the respective regression dependence measures r2|1(X,Y), r1|2(X,Y) ∈ [0,1] introduced in Dette et al. (2013) satisfy r2|1(X,Y) = 1 as well as r1|2(X,Y) <e .