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Extreme values of the mass distribution associated with a tetravariate quasi-copula

Manuel Úbeda-Flores

TL;DR

The paper addresses the problem of determining extreme values of the mass distribution $V_Q$ associated with tetravariate quasi-copulas, extending known results from the bivariate and trivariate cases. It adopts a linear programming framework that enforces boundary, Lipschitz, and monotonicity constraints to locate the minima and maxima of $V_Q(B)$ over $4$-boxes, revealing sharp bounds $-9/7$ and $2$. The extremal configurations are realized by symmetric boxes $B_1=[3/7,6/7]^4$ and $B_2=[1/2,1]^4$, with explicit vertex-value constructions and two example quasi-copulas $Q_1$ and $Q_2$ achieving these masses. The results highlight nontrivial differences across dimensions, discuss issues of non-uniqueness in the extremal solutions, and propose a conjecture for general $n>4$, suggesting further avenues for proving extremal behavior with alternative techniques.

Abstract

In this note we study the extremes of the mass distribution associated with a tetravariate quasi-copula and compare our results with the bi- and trivariate cases, showing the important differences between them.

Extreme values of the mass distribution associated with a tetravariate quasi-copula

TL;DR

The paper addresses the problem of determining extreme values of the mass distribution associated with tetravariate quasi-copulas, extending known results from the bivariate and trivariate cases. It adopts a linear programming framework that enforces boundary, Lipschitz, and monotonicity constraints to locate the minima and maxima of over -boxes, revealing sharp bounds and . The extremal configurations are realized by symmetric boxes and , with explicit vertex-value constructions and two example quasi-copulas and achieving these masses. The results highlight nontrivial differences across dimensions, discuss issues of non-uniqueness in the extremal solutions, and propose a conjecture for general , suggesting further avenues for proving extremal behavior with alternative techniques.

Abstract

In this note we study the extremes of the mass distribution associated with a tetravariate quasi-copula and compare our results with the bi- and trivariate cases, showing the important differences between them.
Paper Structure (2 sections, 1 theorem, 1 equation, 1 figure)

This paper contains 2 sections, 1 theorem, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. The mass distribution associated with a 4-quasi-copula

Key Result

Theorem 1

Let $Q$ be a $4$-quasi-copula and let $B$ be a $4$-box in $[0,1]^4$. Then it holds $-9/7\le V_Q(B)\le 2$.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Example 1