A study on LTD and RTI positive dependence orderings

In this paper we discuss the properties of the orderings of positive dependence introduced by Hollander et al. [Hollander, M., Proschan, F., Sconing, J., 1990. Information, censoring, and dependence. In: Topics in Statistical Dependence (Somerset, PA, 1987). In: IMS Lecture Notes Monogr. Ser., vol. 16. pp. 257-268. Inst. Math. Statist., Hayward, CA] as generalizations of the bivariate positive dependence concepts of left-tail decreasing (LTD) and right-tail increasing (RTI) random vectors studied by Esary and Proschan [Esary, J.D., Proschan, F., 1972. Relationships among some concepts of bivariate dependence. Ann. Math. Statist. 43, 651-655]. We show which of the postulates proposed by Kimeldorf and Sampson [Kimeldorf, G., Sampson, A.R., 1987. Positive dependence orderings. Ann. Inst. Statist. Math. 39 (1), 113-128] for a reasonable positive dependence ordering are satisfied and how the orders can be studied by restricting them to copulas. We also investigate the relationships of these orders with some other orderings which have appeared in the literature and generalize the same notions of positive dependence. Finally, some applications to extreme value bivariate distributions are discussed.