Extremes in High Dimensions: Methods and Scalable Algorithms

Abstract

Extreme-value theory has been explored in considerable detail for univariate and low-dimensional observations, but the field is still in an early stage regarding high-dimensional multivariate observations. In this paper, we focus on Hüsler-Reiss models and their domain of attraction, a popular class of models for multivariate extremes that exhibit some similarities to multivariate Gaussian distributions. We devise novel estimators for the parameters of this model based on score matching and equip these estimators with state-ofthe-art theories for high-dimensional settings and with exceptionally scalable algorithms. We perform a simulation study to demonstrate that the estimators can estimate a large number of parameters reliably and fast; for example, we show that Hüsler-Reiss models with thousands of parameters can be fitted within a couple of minutes on a standard laptop.

On pairwise interaction multivariate Pareto models

“…Remark 3. Part ðiÞ in Lemma 2 was already stated and proved by Lederer and Oesting (2023) and is in fact the main justification for working with F gen HR . The contribution here is in giving a complete picture of the functions in this class.…”

“…As mentioned after the statement of the result, ðiÞ is already obtained by Lederer and Oesting (2023), so only ðiiÞ and ðiiiÞ shall be proved here.…”

“…The Hüsler-Reiss family has been the focus of many recent papers on high-dimensional modeling and inference for tail dependence, especially in relation to extremal graphical models (see, e.g., Asenova et al, 2021;Hentschel et al, 2022;Lederer & Oesting, 2023;Röttger, Coons, & Grosdos, 2023;Röttger, Engelke, & Zwiernik, 2023). In fact, it is the only pairwise interaction family of multivariate Pareto distributions that can be found in the literature, despite the fact that such families are naturally related to the density factorization property underlying extremal graphical models.…”

“…In particular, in order to develop an efficient score matching algorithm, Lederer and Oesting (2023) introduce the class of functions…”