Geometrical representation and dependence structure of three-dimensional Bernoulli distributions

TL;DR

This paper characterizes the geometric structure of three-dimensional Bernoulli distributions with equal means, providing algebraic representations, extremal dependence analysis, and an application to risk minimization using game theory.

Contribution

It offers a complete geometric and algebraic characterization of 3D Bernoulli distributions with equal means, including extremal dependence analysis and a risk application.

Findings

Closed-form geometric generators as functions of p

Algebraic representation encoding Bernoulli properties

Impact of negative dependence on minimal risk

Abstract

This paper fully characterizes the geometrical structure of the class of distributions of three-dimensional Bernoulli random variables with equal means, $p$. We find all the geometrical generators in closed form as functions of $p$. This result stems from an algebraic representation of the class that encodes the statistical properties of Bernoulli distributions. We study extremal negative dependence within the class and provide an application example by finding the impact of negative dependence to minimal aggregate risk. The application relies on a game theory approach.