Orderings for Positive Dependence on Multivariate Empirical Distributions

Abstract

The study of orderings for positive dependence on bivariate empirical distributions can be viewed as the study of partial orderings on the set $S_N$ of all permutations of the integers $1,\ldots,N$. This paper extends earlier bivariate results to multivariate empirical distributions, with focus on the trivariate case. In terms of a newly defined notion of relative rearrangement, characterizations are given of the more positively upper orthant dependent ordering and related orderings. A new partial ordering describing concordance on $(S_N)^m$ is also introduced and connected with the positively upper orthant dependence ordering.