Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal signed volume for (multivariate) supermodular quasi-copulas

Matjaž Omladič, Martin Vuk, Aljaž Zalar

Abstract

Copulas are the primary tool for dependence modeling in statistics, and quasi-copulas are their essential companions. The latter appear, say, as infima or suprema of sets of copulas; they form a huge class and have some unpleasant properties. Their statistical interpretation is challenged by the fact that they may lead to negative volumes of some boxes. So, numerous applications call for an intermediate class, and supermodular quasi-copulas are one of them, having many useful properties. An excellent measure, Average Rectangular Volume (ARV in short), to clarify and position this class was proposed in the seminal paper by Anzilli and Durante, The average rectangular volume induced by supermodular aggregation functions, J. Math. Anal. Appl. 555 (2026) 21 pp. While supermodularity is a bivariate notion, its extension to the $d$-variate case for $d>2$ was recently emphasized in a key paper by Arias-Garcia, Mesiar, and De Baets, The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions, Int. J. of Appr. Reasoning, 80 (2017) pp. 89-99. Here, an alternative method to ARV is presented, extendable to the multivariate case based on Maximal (in absolute value) Negative Volumes (MNV in short) on boxes, thus helping practitioners when seeking the right (quasi-)copula for their problem. Observe that these volumes on copulas are zero, while their values on quasi-copulas, depending on $d$, have been a long-standing open problem solved only recently. We present a nontrivial extension of this solution, which serves as the main goal of this paper: a measure that clarifies and positions the classes considered based on MNV.

Maximal signed volume for (multivariate) supermodular quasi-copulas

Abstract

Copulas are the primary tool for dependence modeling in statistics, and quasi-copulas are their essential companions. The latter appear, say, as infima or suprema of sets of copulas; they form a huge class and have some unpleasant properties. Their statistical interpretation is challenged by the fact that they may lead to negative volumes of some boxes. So, numerous applications call for an intermediate class, and supermodular quasi-copulas are one of them, having many useful properties. An excellent measure, Average Rectangular Volume (ARV in short), to clarify and position this class was proposed in the seminal paper by Anzilli and Durante, The average rectangular volume induced by supermodular aggregation functions, J. Math. Anal. Appl. 555 (2026) 21 pp. While supermodularity is a bivariate notion, its extension to the -variate case for was recently emphasized in a key paper by Arias-Garcia, Mesiar, and De Baets, The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions, Int. J. of Appr. Reasoning, 80 (2017) pp. 89-99. Here, an alternative method to ARV is presented, extendable to the multivariate case based on Maximal (in absolute value) Negative Volumes (MNV in short) on boxes, thus helping practitioners when seeking the right (quasi-)copula for their problem. Observe that these volumes on copulas are zero, while their values on quasi-copulas, depending on , have been a long-standing open problem solved only recently. We present a nontrivial extension of this solution, which serves as the main goal of this paper: a measure that clarifies and positions the classes considered based on MNV.

Paper Structure

This paper contains 5 sections, 9 theorems, 63 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Statements of the main theorems
  3. Towards the proof of Theorems \ref{['sol:general-increasing']} and \ref{['sol:general-increasing-maximal']}
  4. Proofs of Theorems \ref{['sol:general-increasing']} and \ref{['sol:general-increasing-maximal']}
  5. Concluding remarks and future research

Key Result

Theorem 1

Assume the notation above. Let $d,k\in \mathbb N$ and $2\leq k\leq d$. Then where $\gamma_i^{(j)}$ are defined recursively by Let $i_0\in \{k,\ldots,d\}$ be such that One of the realizations of $Q$ and ${\mathcal{B}}$ such that $V_Q({\mathcal{B}}) = \mathrm{MNV}(d,k)$ has the following properties: where $q_i$ are defined as follows and $\alpha_i^{(j)}$ are defined recursively by This realiza

Figures (1)

  • Figure 1: To construct a $k$-increasing 3-quasi-copula $Q$ given the values $q_\mathbb{I}$ at $x_\mathbb{I}\in \mathcal{D}:=\prod_{i=1}^3\{a_i,1\}$, we first define it to be 0 on all $2$-dimensional faces $\mathcal{L}_1$, $\mathcal{L}_2,$$\mathcal{L}_3$ and prove that the extension indeed meets the requirements of a $k$-increasing quasi-copula.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • proof
  • proof : Proof of Proposition \ref{['prel:k-prop']}
  • Proposition 2
  • proof
  • Corollary 1
  • Lemma 2
  • ...and 7 more