Functional Characterizations of Some Positively Dependent Bivariate Random Vectors

In this paper, we consider three different notions of positive dependence for a bivariate random vector: (i) total positivity of order 2, (ii) stochastic increasingness, and (iii) positive quadrant dependence. By defining three classes of arrangement-increasing functions, we show that these three different notions can be unified by functional inequalities. Using these functional inequalities, we derive the relations between the positively dependent notions (i) and (iii) and their corresponding counterparts in stochastic majorization orderings. Moreover, these classes of functions also lead to equivalent characterizations of two random vectors with independent components ordered in the sense of likelihood ratio ordering and stochastic ordering.