Directional variograms for multivariate extremes

2026-07-03

AI Summary

This study addresses the inefficiency of variogram estimation in traditional multivariate extreme-value models that condition on a single component. To overcome this limitation, the authors propose a directional extremal variogram framework by conditioning the multivariate generalized Pareto vector on an arbitrary half-space, introducing a direction vector \( v \) to define the \( v \)-variogram and establishing its decomposition structure. Analytical expressions are derived for the Logistic, Dirichlet, and Hüsler–Reiss models, revealing a unique optimal direction \( v_0 \) in the Hüsler–Reiss case—linked to a resistance–curvature structure—that governs the bias–variance trade-off. Empirical estimation combining multiple directions substantially reduces variance and enhances statistical efficiency.

Abstract

Multivariate generalized Pareto distributions arise as limits of threshold exceedances and form a central model class for multivariate extremes. Existing inference methods based on the extremal variogram condition on the value of a single component, which can be statistically suboptimal. We generalize this approach by conditioning the multivariate generalized Pareto random vector $Y$ to lie on arbitrary half-spaces. Specifically, for a direction vector $v$, we introduce the random vector $Y^v = (Y \mid v^\top Y > 0)$ and define the associated $v$-variogram $Γ_{ij}^v=\mathrm{Var}(Y_i^v-Y_j^v)$. We establish the decomposition $Y^v \stackrel{d}{=} W^v+E\mathbf{1}$ into the so-called $v$-extremal function $W^v$ and an independent exponential random variable $E$, and derive several results relating these random variables to each other. For logistic, Dirichlet, and Hüsler-Reiss multivariate generalized Pareto models, we derive closed-form expressions for $Γ^v$. In the Hüsler-Reiss case, we further derive new density representations and identify a distinguished resistance-curvature vector $v_0$ that uniquely centers the Gaussian law of $W^{v_0}$ while characterizing the least-mass half-space. On the statistical side, we introduce empirical $v$-variograms and show in a simulation study that the choice of $v$ induces a pronounced bias-variance trade-off that is strongly related to the mass of the conditioning half-space. Moreover, combining information across multiple directions $v$ can substantially reduce estimation variance relative to methods based on a single vector.

Problem

• multivariate extremes
• generalized Pareto distributions
• extremal variogram
• directional conditioning
• statistical inference

Innovation

• directional variogram
• multivariate generalized Pareto distribution
• extremal function
• Hüsler–Reiss model
• bias-variance trade-off