Some remarks on the supermodular order

In this paper we solve two open problems posed by Joe (1997) concerning the supermodular order. First we give an example which shows that the supermod ular order is strictly stronger than the concordance order for dimension d = 3. Second we show that the supermodular order fulfils all desirable prope rties of a multivariate positive dependence order. We especially prove the non-trivial fact that it is closed with respect to weak convergence. This i s applied to give a complete characterization of the supermodular order for multivariate normal distributions. (C) 2000 Academic Press.