Extremal negative dependence and the strongly Rayleigh property
Hélène Cossette, Etienne Marceau, Alessandro Mutti, Patrizia Semeraro
TL;DR
The paper investigates extremal negative dependence for multidimensional Bernoulli distributions by focusing on $\Sigma$-countermonotonicity within the Fréchet class $\mathcal{B}_d(\boldsymbol{p})$. It proves a geometric characterization: $\mathcal{B}_d^{\Sigma}(\boldsymbol{p})$ is a convex polytope whose extremal points are $\Sigma$-countermonotonic, and it establishes the equivalence of $\Sigma$-countermonotonicity with $\Sigma_{cx}$-minimality. The authors show that, despite the class not guaranteeing the strongly Rayleigh property, SR distributions exist in $\mathcal{B}_d^{\Sigma}(\boldsymbol{p})$ and the maximum-entropy member is SR and can be represented as a conditional Bernoulli distribution; they also demonstrate a minimality property of the class with respect to weak association and supermodular orders, including an antichain structure and a chain to independence via convex mixtures. Collectively, these results clarify how extremal negative dependence interacts with the SR property and dependence orders, informing both theory and applications involving joint Bernoulli distributions and negative dependence modeling.
Abstract
We provide a geometrical characterization of extremal negative dependence as a convex polytope in the simplex of multidimensional Bernoulli distributions, and we prove that it is an antichain that satisfies some minimality conditions with respect to the strongest negative dependence orders. We study the strongly Rayleigh property within this class and explicitly find a distribution that satisfies this property by maximizing the entropy. Furthermore, we construct a chain for the supermodular order starting from extremal negative dependence to independence by mixing the maximum entropy strongly Rayleigh distribution with independence.