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.

翻译：单变量与低维观测的极值理论已得到相当深入的探索，但该领域在高维设定方面仍处于早期阶段。本文聚焦于Hüsler-Reiss模型——一类类似于多元高斯分布的流行多元极值模型及其吸引域。我们基于得分匹配开发了模型参数的估计器，并为这些估计器配备了理论支撑与高度可扩展的算法。针对极端天气的模拟与应用表明，这些估计器能够快速可靠地估计大量参数；例如，我们证明包含数千参数的Hüsler-Reiss模型可在标准笔记本电脑上数分钟内完成拟合。更广义而言，我们的工作将极值理论与高维统计和凸优化的现代概念联系起来。