Extreme values of the mass distribution associated with $d$-quasi-copulas via linear programming

TL;DR

This paper investigates the extremal volume bounds of multivariate quasi-copulas, disproves a recent conjecture on lower bounds, and highlights the complexity of the problem using linear programming techniques.

Contribution

It disproves a conjecture on the lower volume bound of d-quasi-copulas and provides insights into the problem's complexity.

Findings

Disproved a recent conjecture on lower bounds.

Indicated the problem's difficulty exceeds initial expectations.

Suggested directions for future research.

Abstract

The recent survey published in Fuzzy Sets and Systems nicknamed ``Hitchhiker's Guide'' has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. Some of the open problems listed there were solved, and some conjectured one way or the other. This paper concentrates on the Open Problem 5 of this list concerning bounds on the volume of a $d$--variate quasi-copula. We disprove a recent conjecture published in the same journal on the lower bound of this volume. We also give evidence that the problem is much more difficult than suspected and provide hints about its final solution.