Statistical inference for multivariate extremes via a geometric approach

TL;DR

This paper introduces a geometric approach for statistical inference on multivariate extremes, enabling flexible modeling and accurate tail probability estimation through parametric and semi-parametric methods, validated by simulations and environmental data applications.

Contribution

It develops a new geometric-based statistical inference method for multivariate extremes, including a parametric estimation of limit set shapes and models for tail dependence.

Findings

Method is competitive with existing approaches.

Successfully estimates small tail probabilities.

Demonstrates good fit on environmental datasets.

Abstract

A geometric representation for multivariate extremes, based on the shapes of scaled sample clouds in light-tailed margins and their so-called limit sets, has recently been shown to connect several existing extremal dependence concepts. However, these results are purely probabilistic, and the geometric approach itself has not been fully exploited for statistical inference. We outline a method for parametric estimation of the limit set shape, which includes a useful non/semi-parametric estimate as a pre-processing step. More fundamentally, our approach provides a new class of asymptotically-motivated statistical models for the tails of multivariate distributions, and such models can accommodate any combination of simultaneous or non-simultaneous extremes through appropriate parametric forms for the limit set shape. Extrapolation further into the tail of the distribution is possible via simulation from the fitted model. A simulation study confirms that our methodology is very competitive with existing approaches, and can successfully allow estimation of small probabilities in regions where other methods struggle. We apply the methodology to two environmental datasets, with diagnostics demonstrating a good fit.