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Freedom in constructing quasi-copulas vs. copulas

Matjaž Omladič, Nik Stopar

TL;DR

The paper investigates the freedom to construct quasi-copulas versus copulas, showing that every $d$-variate quasi-copula ($d\ge 3$) can be obtained by two-step pointwise infimum/supremum operations on sets of copulas, with a bivariate one-step version. It extends the powerful shuffle-of-min technique to quasi-copulas, proving that any quasi-copula can be uniformly approximated by a shuffle of a suitable quasi-copula, and provides a multivariate tensor-product framework that preserves quasi-copula structure while enabling mass-preserving composition. A novel, constructive method prescribes how to realize any signed mass pattern on a patch as a discrete quasi-copula after appropriate scaling, demonstrating substantial local flexibility. Collectively, these results illuminate the richer landscape of quasi-copulas relative to copulas, while clarifying the limits and opportunities of shuffling and mass-pattern constructions, with implications for imprecise copulas and Hitchhiker-type problems. The work advances practical construction tools and theoretical understanding of the relationship between quasi-copulas and copulas.

Abstract

The main goal of this paper is to study the extent of freedom one has in constructing quasi-copulas vs. copulas. Specifically, it exhibits three construction methods for quasi-copulas based on recent developments: a representation of multivariate quasi-copulas by means of infima and suprema of copulas, an extension of a classical result on shuffles of min to the setting of quasi-copulas, and a construction method for quasi-copulas obeying a given signed mass pattern on a patch.

Freedom in constructing quasi-copulas vs. copulas

TL;DR

The paper investigates the freedom to construct quasi-copulas versus copulas, showing that every -variate quasi-copula () can be obtained by two-step pointwise infimum/supremum operations on sets of copulas, with a bivariate one-step version. It extends the powerful shuffle-of-min technique to quasi-copulas, proving that any quasi-copula can be uniformly approximated by a shuffle of a suitable quasi-copula, and provides a multivariate tensor-product framework that preserves quasi-copula structure while enabling mass-preserving composition. A novel, constructive method prescribes how to realize any signed mass pattern on a patch as a discrete quasi-copula after appropriate scaling, demonstrating substantial local flexibility. Collectively, these results illuminate the richer landscape of quasi-copulas relative to copulas, while clarifying the limits and opportunities of shuffling and mass-pattern constructions, with implications for imprecise copulas and Hitchhiker-type problems. The work advances practical construction tools and theoretical understanding of the relationship between quasi-copulas and copulas.

Abstract

The main goal of this paper is to study the extent of freedom one has in constructing quasi-copulas vs. copulas. Specifically, it exhibits three construction methods for quasi-copulas based on recent developments: a representation of multivariate quasi-copulas by means of infima and suprema of copulas, an extension of a classical result on shuffles of min to the setting of quasi-copulas, and a construction method for quasi-copulas obeying a given signed mass pattern on a patch.
Paper Structure (6 sections, 13 theorems, 55 equations, 3 figures)

This paper contains 6 sections, 13 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Lattice-theoretic properties of multivariate quasi-copulas
  3. Shuffles of quasi-copulas and bivariate approximation
  4. Multivariate approximation
  5. Quasi-copula obeying a given signed mass pattern on a patch
  6. Concluding remarks

Key Result

Proposition 1

Let $\mathcal{D}$ be a family of copulas such that for every quasi-copula $Q$ and every two points $\mathbf{x},\mathbf{z}\in\mathds{I}^d$ there exists $D_{\mathbf{x},\mathbf{z}}^Q\in\mathcal{D}$ such that $D_{\mathbf{x},\mathbf{z}}^Q(\mathbf{x})=Q(\mathbf{x})$ and $D_{\mathbf{x},\mathbf{z}}^Q(\mathb for all $\mathbf{u} \in \mathds{I}^d$.

Figures (3)

  • Figure 1: Mass distributions of copulas $C_1,C_2,C_3,C_4$ and their shuffles from Example \ref{['ex:sh1']}, along with the mass distributions of some of their supremums. The full lines indicate positive mass while the dotted line indicates negative mass.
  • Figure 2: Mass distribution of an example quasi-copulas $Q$ (left) and corresponding quasi-copulas $B$ (middle) and $A$ (right) from the proof of Theorem \ref{['thm:bishuffle']} for $m=4$. On each big square quasi-copula $Q$ spreads either mass $\frac{1}{4}$ uniformly on one of the diagonals, or mass $-\frac{1}{4}$ uniformly on the square, or mass $0$.
  • Figure 3: The mass distribution of a quasi-copula $Q$ that is an ordinal sum of $3$ quasi-copulas (left) and the mass distribution of the corresponding two-way shuffle $Q^{(S,S)}$ (right) induced by the permutation $\pi=123456789417586293$. The total mass in the stripped region on the right equals $Q^{(S,S)}(u,v)$ and the relative positions (within the respective squares) of the corresponding points $u_i$ and $v_i$ from the proof of Theorem \ref{['thm:stochastic']} are depicted on the left.

Theorems & Definitions (29)

  • Proposition 1
  • proof
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Example 6
  • Proposition 7
  • ...and 19 more