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Extreme semilinear copulas

Fabrizio Durante, Juan Fernández-Sánchez, Manuel Úbeda-Flores

TL;DR

This work characterizes the extreme points of the convex class of semilinear copulas through their diagonal sections: a semilinear copula $C_{\delta}$ is extreme iff its diagonal $\delta$ is extreme in the diagonal set $D_{C_S}$. The authors establish a homeomorphism between the semilinear copulas and their diagonals, apply Krein–Milman and Choquet theory to obtain representations, and illustrate with a tractable subclass $\delta_m(t)=(mt)\vee t^2$ that yields explicit association measures and a dense hull of extreme elements. They also analyze asymmetry maps, showing that extremal asymmetries are realized by these extreme diagonals, and extend the extremality framework to semilinear semi--copulas and quasi--copulas, highlighting richer extremal structures and noncopula extreme quasi--copulas. Collectively, the results provide a diagonal-based blueprint for understanding and representing extremal semilinear copulas and their generalized aggregation counterparts, with implications for tail behavior and symmetry analyses.

Abstract

We study the extreme points (in the Krein-Milman sense) of the class of semilinear copulas and provide their characterization. Related results into the more general setting of conjunctive aggregation functions (i.e, semi--copulas and quasi--copulas) are also presented.

Extreme semilinear copulas

TL;DR

This work characterizes the extreme points of the convex class of semilinear copulas through their diagonal sections: a semilinear copula is extreme iff its diagonal is extreme in the diagonal set . The authors establish a homeomorphism between the semilinear copulas and their diagonals, apply Krein–Milman and Choquet theory to obtain representations, and illustrate with a tractable subclass that yields explicit association measures and a dense hull of extreme elements. They also analyze asymmetry maps, showing that extremal asymmetries are realized by these extreme diagonals, and extend the extremality framework to semilinear semi--copulas and quasi--copulas, highlighting richer extremal structures and noncopula extreme quasi--copulas. Collectively, the results provide a diagonal-based blueprint for understanding and representing extremal semilinear copulas and their generalized aggregation counterparts, with implications for tail behavior and symmetry analyses.

Abstract

We study the extreme points (in the Krein-Milman sense) of the class of semilinear copulas and provide their characterization. Related results into the more general setting of conjunctive aggregation functions (i.e, semi--copulas and quasi--copulas) are also presented.
Paper Structure (7 sections, 19 theorems, 68 equations, 3 figures)

This paper contains 7 sections, 19 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extreme semilinear copulas
  4. A subclass of extreme semilinear copulas
  5. Asymmetry maps of semilinear copulas
  6. Extreme semilinear semi-copulas and quasi-copulas
  7. Conclusions

Key Result

Theorem 1

The function given by eq:semilinear is a lower semilinear copula if, and only if, the functions $x\mapsto \varphi_\delta(x):=\delta(x)/x$ and $x\mapsto \eta_\delta(x):=\delta(x)/x^2$ are non-decreasing and non-increasing, respectively, on $]0,1]$.

Figures (3)

  • Figure 1: Support of the extreme semilinear copulas generated by \ref{['eq:deltam']}.
  • Figure 2: The bounds $L$ (left) and $U$ (right) of Theorem \ref{['T:oppositediag']}.
  • Figure 3: The map of radial asymmetry of Theorem \ref{['T:propradial']}.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4: Krein-Milman
  • Theorem 5: Choquet
  • Corollary 6
  • proof
  • Corollary 7
  • ...and 17 more