Almost opposite regression dependence in bivariate distributions

Let $$X$$ X , $$Y$$ Y be two continuous random variables. Investigating the regression dependence of $$Y$$ Y on $$X$$ X , respectively, of $$X$$ X on $$Y$$ Y , we show that the two of them can have almost opposite behavior. Indeed, given any $$\epsilon >0$$ ϵ > 0 , we construct a bivariate random vector $$(X,Y)$$ ( X , Y ) such that the respective regression dependence measures $$r_{2|1}(X,Y), r_{1|2}(X,Y) \in [0,1]$$ r 2 | 1 ( X , Y ) , r 1 | 2 ( X , Y ) ∈ [ 0 , 1 ] introduced in Dette et al. (Scand. J. Stat. 40(1):21–41, 2013 ) satisfy $$r_{2|1}(X,Y)=1$$ r 2 | 1 ( X , Y ) = 1 as well as $$r_{1|2}(X,Y) > \epsilon $$ r 1 | 2 ( X , Y ) > ϵ . Copyright Springer-Verlag Berlin Heidelberg 2015