Partial Orders on Permutations and Dependence Orderings on Bivariate Empirical Distributions

Abstract

Three well-known partial orderings and one new one, designated by $\geq_{b_t}, t = 1, 2, 3, 4$, are defined on permutations of $\{1, 2, \cdots, n\}$ through a unified approach. Various formulations of these partial orderings are also considered. With the aid of these formulations we show that the four orderings on permutations are equivalent to positive dependence orderings defined over empirical distributions based on rank data. In particular, we show that the orderings $b_1, b_2, b_3$ and $b_4$ are equivalent, respectively, to more concordant, more row regression dependent, more column regression dependent and more associated.