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 $\epsilon >0$, we construct a bivariate random vector $(X,Y)$ such that the respective regression dependence measures $r_{2|1}(X,Y), r_{1|2}(X,Y) \in [0,1]$ introduced in Dette et al. (Scand. J. Stat. 40(1):21–41, 2013) satisfy $r_{2|1}(X,Y) = 1$ as well as $r_{1|2}(X,Y) < \epsilon$.