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Extremal conditional independence for Hüsler-Reiss distributions via modular functions

Karel Devriendt, Ignacio Echave-Sustaeta Rodríguez, Frank Röttger

Abstract

We study extremal conditional independence for Hüsler-Reiss distributions, which is a parametric subclass of multivariate Pareto distributions. As the main contribution, we introduce two set functions, i.e.~functions which assign a value to the distribution and each of its marginals, and show that extremal conditional independence statements can be characterized by modularity relations for these functions. For the first function, we make use of the close connection between Hüsler-Reiss and Gaussian models to introduce a multiinformation-inspired measure $m^{\text{HR}}$ for Hüsler-Reiss distributions. For the second function, we consider an invariant $σ^2$ that is naturally associated to the Hüsler-Reiss parameterization and establish the second modularity criterion under additional positivity constraints. Together, these results provide new tools for describing extremal dependence structures in high-dimensional extreme value statistics. In addition, we study the geometry of a bounded subset of Hüsler-Reiss parameters and its relation with the Gaussian elliptope.

Extremal conditional independence for Hüsler-Reiss distributions via modular functions

Abstract

We study extremal conditional independence for Hüsler-Reiss distributions, which is a parametric subclass of multivariate Pareto distributions. As the main contribution, we introduce two set functions, i.e.~functions which assign a value to the distribution and each of its marginals, and show that extremal conditional independence statements can be characterized by modularity relations for these functions. For the first function, we make use of the close connection between Hüsler-Reiss and Gaussian models to introduce a multiinformation-inspired measure for Hüsler-Reiss distributions. For the second function, we consider an invariant that is naturally associated to the Hüsler-Reiss parameterization and establish the second modularity criterion under additional positivity constraints. Together, these results provide new tools for describing extremal dependence structures in high-dimensional extreme value statistics. In addition, we study the geometry of a bounded subset of Hüsler-Reiss parameters and its relation with the Gaussian elliptope.
Paper Structure (10 sections, 13 theorems, 56 equations, 2 figures)

This paper contains 10 sections, 13 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multivariate extremes and extremal conditional independence
  4. Hüsler--Reiss distributions
  5. Extremal conditional independence via Gaussian multiinformation
  6. Extremal conditional independence for Hüsler--Reiss distributions
  7. Extremal conditional independence via the resistance radius
  8. Marginalization and properties of the resistance curvature and resistance radius
  9. Extremal conditional independence
  10. Geometry of variogram matrices

Key Result

Lemma 2.1

Let $\mathbf{Y}$ be a Hüsler--Reiss vector with variogram $\Gamma$ and precision matrix $\Theta$. Then for any nonempty disjoint subsets $\{i\},\{j\},C\subset[d]$, the following are equivalent:

Figures (2)

  • Figure 1: The $3$-dimensional Hüsler--Reiss elliptope $\mathcal{F}_3\subset\mathbb{R}^3$. The red lines correspond to correlation matrices removed as described in Proposition \ref{['PROP:ELLIPTOPE']}.
  • Figure 2: The Hüsler--Reiss elliptope $\mathcal{F}_3$ intersected with the region where $\mathbf{p} \geq 0$ holds or, equivalently for $d=3$, where $\text{EMTP}_2$ holds.

Theorems & Definitions (36)

  • Definition 1: engelkehitz
  • Lemma 2.1: EGR2025, RCG2023
  • Example 1: Four-cycle
  • Definition 2: Relative entropy
  • Definition 3: Multiinformation
  • Theorem 3.1: Studeny_2005
  • Example 2: Gaussian distributions
  • Definition 4
  • Proposition 3.2
  • proof
  • ...and 26 more