Joint mixability and negative orthant dependence

A joint mix is a random vector with a constant component-wise sum. It is known to represent the minimizing dependence structure of some common objectives, and it is usually regarded as a concept of extremal negative dependence. In this paper, we explore the connection between the joint mix structure and one of the most popular notions of negative dependence in statistics, called negative orthant dependence (NOD). We show that a joint mix does not always have NOD, but some natural classes of joint mixes have. In particular, the Gaussian class is characterized as the only elliptical class which supports NOD joint mixes of arbitrary dimension. For Gaussian margins, we also derive a necessary and sufficient condition for the existence of an NOD joint mix. Finally, for identical marginal distributions, we show that an NOD Gaussian joint mix solves a multi-marginal optimal transport problem under uncertainty on the number of components. Analysis of this optimal transport problem with heterogeneous marginals reveals a trade-off between NOD and the joint mix structure.