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Characterizing extremal dependence on a hyperplane

Phyllis Wan

TL;DR

The paper reframes multivariate extremes by projecting tail dependence onto the hyperplane perpendicular to the diagonal, turning nonlinear tail dependencies into a linear structure that supports linear techniques like PCA. It introduces diagonal peak-over-threshold and profile random vectors, linking diagonal GP tails to projections of spectral vectors and showing tail limits can be expressed as a sum of an Exp(1) term and a profile on $oldsymbol{1}^{\, op}$. A key contribution is the demonstration that Gaussian profile random vectors correspond to the Hüsler-Reiss model, enabling direct generation, dimensionality reduction, and inference for extreme-value dependence in a linear setting. Overall, the framework provides a tractable, geometry-guided approach to modeling and analyzing extremal dependence in high dimensions with practical implications for tail approximations and parametric modeling.

Abstract

In this paper, we characterize the extremal dependence of $d$ asymptotically dependent variables by a class of random vectors on the $(d-1)$-dimensional hyperplane perpendicular to the diagonal vector $\mathbf1=(1,\ldots,1)$. This translates analyses of multivariate extremes to that on a linear vector space, opening up possibilities for applying existing statistical techniques that are based on linear operations. As an example, we demonstrate obtaining lower-dimensional approximations of the tail dependence through principal component analysis. Additionally, we show that the widely used Hüsler-Reiss family is characterized by a Gaussian family residing on the hyperplane.

Characterizing extremal dependence on a hyperplane

TL;DR

The paper reframes multivariate extremes by projecting tail dependence onto the hyperplane perpendicular to the diagonal, turning nonlinear tail dependencies into a linear structure that supports linear techniques like PCA. It introduces diagonal peak-over-threshold and profile random vectors, linking diagonal GP tails to projections of spectral vectors and showing tail limits can be expressed as a sum of an Exp(1) term and a profile on . A key contribution is the demonstration that Gaussian profile random vectors correspond to the Hüsler-Reiss model, enabling direct generation, dimensionality reduction, and inference for extreme-value dependence in a linear setting. Overall, the framework provides a tractable, geometry-guided approach to modeling and analyzing extremal dependence in high dimensions with practical implications for tail approximations and parametric modeling.

Abstract

In this paper, we characterize the extremal dependence of asymptotically dependent variables by a class of random vectors on the -dimensional hyperplane perpendicular to the diagonal vector . This translates analyses of multivariate extremes to that on a linear vector space, opening up possibilities for applying existing statistical techniques that are based on linear operations. As an example, we demonstrate obtaining lower-dimensional approximations of the tail dependence through principal component analysis. Additionally, we show that the widely used Hüsler-Reiss family is characterized by a Gaussian family residing on the hyperplane.

Paper Structure

This paper contains 8 sections, 9 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on multivariate extremes
  3. Diagonal peak-over-threshold and profile random vectors
  4. Diagonal peak-over-threshold
  5. Profile random vectors
  6. Principal component analysis
  7. Gaussian profile random vectors
  8. Proofs

Key Result

Proposition 2.4

Let $\mathcal{S}$ be the class of random vectors $\mathbf{S} \in (-\infty,0]^d$ such that $\textup{pr}\{\max(\mathbf{S})=0\}=1$ and $\textup{pr}(S_k > -\infty)=1$, $k=1,\ldots,d$. Let $\mathbf{Z}$ be a standardized multivariate generalized Pareto distribution with asymptotically dependent components where $\mathbf{S} \in \mathcal{S}$ and $E$ is an $\text{Exp}(1)$ variable independent of $\mathbf{S

Figures (1)

  • Figure 1: Plot (a): Domain of $\mathbf{S}$ (thick line), direction of $E \cdot \mathbf1$ (arrow), and domain of $\mathbf{Z} \overset{d}= \mathbf{S} + E \cdot \mathbf1$ (shaded area); Plot (b): Domain of $\mathbf{U}$ (thick line), direction of $E' \cdot \mathbf1$ (arrow), and domain of $\mathbf{Z}^* \overset{d}= \mathbf{U} + E' \cdot \mathbf1$ (shaded area); Plot (c): Example of projection from $\mathbf{S}$ to $\mathbf{T} = \mathbf{S} - \bar{S} \cdot \mathbf1$; Plot (d): densities of a pair of associated $\mathbf{T}$ and $\mathbf{U}$ on $\mathbf1^\perp$.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • Corollary 3.5
  • Proposition 3.6
  • ...and 13 more