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Joint mixability and notions of negative dependence

Takaaki Koike, Liyuan Lin, Ruodu Wang

TL;DR

The paper investigates how joint mixability (a joint vector with a constant sum) relates to classical negative dependence notions such as NCD, NOD, and NA. It develops both structural results (conditions under which a joint mix is negatively dependent) and optimization insights by formulating a multi-marginal OT problem with uncertainty on component participation; it shows that, for convex costs (notably the quadratic cost), an exchangeable NCD joint mix with correlation matrix $P_n^*$ minimizes the worst-case objective under symmetric marginal constraints, with Gaussian marginals yielding stronger, sometimes unique, optimality results. The elliptical-distribution analysis reveals a sharp dichotomy: NCD JM exists for all dimensions exactly when the marginals form a Gaussian variance mixture, while NA/NOD/NSD across all dimensions characterizes the Gaussian family. These results illuminate the trade-offs between joint-mix structure and various negative-dependent properties, and they hint at broad applicability to risk management, transport, and related optimization problems under dependence uncertainty.

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.

Joint mixability and notions of negative dependence

TL;DR

The paper investigates how joint mixability (a joint vector with a constant sum) relates to classical negative dependence notions such as NCD, NOD, and NA. It develops both structural results (conditions under which a joint mix is negatively dependent) and optimization insights by formulating a multi-marginal OT problem with uncertainty on component participation; it shows that, for convex costs (notably the quadratic cost), an exchangeable NCD joint mix with correlation matrix minimizes the worst-case objective under symmetric marginal constraints, with Gaussian marginals yielding stronger, sometimes unique, optimality results. The elliptical-distribution analysis reveals a sharp dichotomy: NCD JM exists for all dimensions exactly when the marginals form a Gaussian variance mixture, while NA/NOD/NSD across all dimensions characterizes the Gaussian family. These results illuminate the trade-offs between joint-mix structure and various negative-dependent properties, and they hint at broad applicability to risk management, transport, and related optimization problems under dependence uncertainty.

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.
Paper Structure (10 sections, 16 theorems, 70 equations)

This paper contains 10 sections, 16 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Notions of negative dependence
  3. JM and negative dependence
  4. A multi-marginal optimal transport problem
  5. Optimal transport under uncertainty on the set of components
  6. Quadratic cost
  7. Discussions on heterogeneous marginals
  8. Elliptical distributions
  9. Conclusion
  10. Proofs of all results

Key Result

Proposition 1

Let $\mathbf X\sim \operatorname{N}_n(\boldsymbol{\mu},\Sigma)$ be a Gaussian random vector with mean vector $\boldsymbol{\mu}\in \mathbb{R}^n$ and covariance matrix $\Sigma=(\sigma_{ij})_{n\times n}$.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Theorem 1
  • Example 1
  • Example 2
  • Proposition 2
  • Proposition 3
  • Theorem 2
  • Example 3
  • ...and 36 more