Symmetric Bernoulli distributions and minimal dependence copulas

TL;DR

This paper characterizes multivariate symmetric Bernoulli distributions with minimal convex sums, revealing extremal negative dependence structures and constructing specific classes of minimal dependence copulas, thus advancing understanding of negative dependence in copula theory.

Contribution

It provides a complete characterization of symmetric Bernoulli distributions with minimal convex sums and introduces new classes of minimal dependence copulas based on these distributions.

Findings

Identified all symmetric Bernoulli distributions with minimal convex sums.

Constructed extremal negative dependence copulas from these distributions.

Provided geometric and algebraic representations of the distributions.

Abstract

The key result of this paper is to characterize all the multivariate symmetric Bernoulli distributions whose sum is minimal under convex order. In doing so, we automatically characterize extremal negative dependence among Bernoulli random vectors, since multivariate distributions with minimal convex sums are known to be strongly negative dependent. Moreover, beyond its interest per se, this result provides insight into negative dependence within the class of copulas. In particular, two classes of copulas can be built from multivariate symmetric Bernoulli distributions: extremal mixture copulas and FGM copulas. We analyze the extremal negative dependence structures of copulas corresponding to symmetric Bernoulli random vectors with minimal convex sums and explicitly find a class of minimal dependence copulas. Our main results derive from the geometric and algebraic representations of multivariate symmetric Bernoulli distributions, which effectively encode key statistical properties.