Extremes in High Dimensions: Methods and Scalable Algorithms

TL;DR

This paper develops scalable estimation methods for high-dimensional extreme value models, specifically H"usler-Reiss models, enabling fast and reliable analysis of large parameter sets in multivariate extremes.

Contribution

It introduces score matching estimators and algorithms for high-dimensional H"usler-Reiss models, bridging extreme value theory with high-dimensional statistics and convex optimization.

Findings

Estimators can reliably estimate thousands of parameters.

Algorithms fit models within minutes on standard laptops.

Applications to weather extremes demonstrate practical utility.

Abstract

Extreme value theory for univariate and low-dimensional observations has been explored in considerable detail, but the field is still in an early stage regarding high-dimensional settings. This paper focuses on H\"usler-Reiss models, a popular class of models for multivariate extremes similar to multivariate Gaussian distributions, and their domain of attraction. We develop estimators for the model parameters based on score matching, and we equip these estimators with theories and exceptionally scalable algorithms. Simulations and applications to weather extremes demonstrate the fact that the estimators can estimate a large number of parameters reliably and fast; for example, we show that H\"usler-Reiss models with thousands of parameters can be fitted within a couple of minutes on a standard laptop. More generally speaking, our work relates extreme value theory to modern concepts of high-dimensional statistics and convex optimization.