A new method to construct high-dimensional copulas with Bernoulli and Coxian-2 distributions

TL;DR

This paper introduces a novel high-dimensional copula construction method using mixtures of power functions, based on multivariate Bernoulli and Coxian-2 distributions, enabling flexible modeling of dependence and asymmetries.

Contribution

It presents a new approach to construct high-dimensional copulas with exact dependence measures, extending GFGM copulas via stochastic representations involving Bernoulli and Coxian-2 distributions.

Findings

Constructed a family of copulas with multivariate dependence properties

Derived an asymmetric modified Huang-Kotz FGM copula in the bivariate case

Analyzed the most negative dependence structures in the new class

Abstract

We propose an approach to construct a new family of generalized Farlie-Gumbel-Morgenstern (GFGM) copulas that naturally scales to high dimensions. A GFGM copula can model moderate positive and negative dependence, cover different types of asymmetries, and admits exact expressions for many quantities of interest such as measures of association or risk measures in actuarial science or quantitative risk management. More importantly, this paper presents a new method to construct high-dimensional copulas based on mixtures of power functions, and may be adapted to more general contexts to construct broader families of copulas. We construct a family of copulas through a stochastic representation based on multivariate Bernoulli distributions and Coxian-2 distributions. This paper will cover the construction of a GFGM copula, study its measures of multivariate association and dependence properties. We explain how to sample random vectors from the new family of copulas in high dimensions. Then, we study the bivariate case in detail and find that our construction leads to an asymmetric modified Huang-Kotz FGM copula. Finally, we study the exchangeable case and provide some insights into the most negative dependence structure within this new class of high-dimensional copulas.