Joint mixability and notions of negative dependence

TL;DR

This paper investigates the relationship between joint mixes and various notions of negative dependence, establishing conditions for negative dependence and linking joint mixes to optimal transport problems.

Contribution

It provides necessary and sufficient conditions for joint mixes to exhibit negative dependence and connects these concepts to multi-marginal optimal transport under uncertainty.

Findings

Certain classes of joint mixes are negatively dependent.

Negative dependence conditions depend on marginal distributions.

A joint mix can solve a multi-marginal optimal transport problem.

Abstract

A joint mix is a random vector with a constant component-wise sum. The dependence structure of a joint mix minimizes some common objectives such as the variance of the component-wise sum, and it is regarded as a concept of extremal negative dependence. In this paper, we explore the connection between the joint mix structure and popular notions of negative dependence in statistics, such as negative correlation dependence, negative orthant dependence and negative association. A joint mix is not always negatively dependent in any of the above senses, but some natural classes of joint mixes are. We derive various necessary and sufficient conditions for a joint mix to be negatively dependent, and study the compatibility of these notions. For identical marginal distributions, we show that a negatively dependent joint mix solves a multi-marginal optimal transport problem for quadratic cost under a novel setting of uncertainty. Analysis of this optimal transport problem with heterogeneous marginals reveals a trade-off between negative dependence and the joint mix structure.