Learning the dependence structure of rare events: a non-asymptotic study

TL;DR

This paper develops non-asymptotic bounds for estimating the stable tail dependence function in multivariate extreme value analysis, providing finite-sample guarantees using VC-type concentration inequalities.

Contribution

It introduces non-asymptotic bounds for the empirical STDF, filling a gap in the theoretical understanding of multivariate extreme event dependence estimation.

Findings

Derived upper bounds with convergence rate O(k^-1/2)

Applied concentration inequalities to low probability regions

Extended tools to classification of extreme data

Abstract

Assessing the probability of occurrence of extreme events is a crucial issue in various fields like finance, insurance, telecommunication or environmental sciences. In a multivariate framework, the tail dependence is characterized by the so-called stable tail dependence function (STDF). Learning this structure is the keystone of multivariate extremes. Although extensive studies have proved consistency and asymptotic normality for the empirical version of the STDF, non-asymptotic bounds are still missing. The main purpose of this paper is to fill this gap. Taking advantage of adapted VC-type concentration inequalities, upper bounds are derived with expected rate of convergence in O(k^-1/2). The concentration tools involved in this analysis rely on a more general study of maximal deviations in low probability regions, and thus directly apply to the classification of extreme data.