Paper: arXiv 2309.17346

Authors: Alessandro Mutti, Patrizia Semeraro

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides strong theoretical contributions by characterizing symmetric Bernoulli distributions with minimal convex sums and connecting them to extremal negative dependence copulas using geometric and algebraic techniques. However, it is entirely theoretical with minimal empirical or quantitative backtesting elements.

Research Flowchart

  flowchart TD
    A[Goal: Characterize Symmetric Bernoulli<br>Distributions with Minimal Convex Sums] --> B[Geometric & Algebraic Representations<br>of Bernoulli Random Vectors]
    B --> C[Derive Extremal Negative Dependence<br>Conditions for Multivariate Bernoulli]
    C --> D[Construct Copula Classes:<br>Extremal Mixture & FGM Copulas]
    D --> E[Analyze Copula Structures<br>under Minimal Convex Sums]
    E --> F[Outcome: Explicit Characterization of<br>Minimal Dependence Copulas]