Extreme mass distributions for quasi-copulas

TL;DR

This paper investigates the extreme mass distributions of multivariate quasi-copulas, providing a complete analytical solution to a previously open problem and advancing the understanding of their dependence structures.

Contribution

It offers an analytical approach to solve the open problem on extreme mass distributions of quasi-copulas, extending previous linear programming results and disproving a recent conjecture.

Findings

Solved the open problem for dimensions up to 17

Disproved a recent conjecture on quasi-copula solutions

Provided a complete analytical characterization of extreme mass distributions

Abstract

A recent survey, nicknamed "Hitchhiker's Guide", J.J. Arias-Garc{\i}a, R. Mesiar, and B. De Baets, A hitchhiker's guide to quasi-copulas, Fuzzy Sets and Systems 393 (2020) 1-28, has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. In our previous work (Fuzzy Sets and Systems 517 (2025) 109457), we addressed the question of extreme values of the mass distribution associated with multivariate quasi-copulas. Using a linear programming approach, we were able to solve Open Problem 5 of the "Guide" up to dimension d = 17 and disprove a recent conjecture on the solution to that problem. In this paper, we use an analytical approach to provide a complete answer to the original question.