[Paper Review] Learning the dependence structure of rare events: a non-asymptotic study

Nicolas Goix, Anne Sabourin+1 more|arXiv (Cornell University)|May 23, 2015

Financial Risk and Volatility Modeling18 references5 citations

TL;DR

This paper provides non-asymptotic concentration bounds for the empirical stable tail dependence function (stdf) in multivariate extreme value theory, using adaptive VC-type inequalities tailored to low-probability regions. It establishes an expected rate of convergence of $O(k^{-1/2})$ without requiring smoothness assumptions, offering finite-sample guarantees for learning dependence structures of rare events.

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.

Motivation & Objective

• To address the lack of finite-sample error bounds for empirical estimators of the stable tail dependence function (stdf) in multivariate extreme value theory.
• To develop concentration inequalities specifically suited for maximal deviations in low-probability regions, relevant to extreme data.
• To derive non-asymptotic convergence rates for the empirical stdf estimator under minimal assumptions, including only the existence of the stdf.
• To enable reliable statistical inference for rare event dependence structures using moderate-order extreme order statistics.
• To support practical classification and risk assessment of extreme multivariate events with finite-sample confidence.

Proposed method

• The authors employ a general framework of maximal deviation bounds over adapted VC classes that cover only low-probability regions corresponding to extreme data.
• They derive VC-type concentration inequalities that explicitly incorporate the probability $p$ of hitting the class, enabling tighter control in rare-event regimes.
• The method relies on a transformation of the data to uniform margins via probability integral transforms, facilitating the use of empirical process tools.
• Theoretical analysis leverages standard multivariate regular variation and the existence of an exponent measure to characterize tail dependence.
• The empirical stdf is estimated using the $k$ largest order statistics from a sample of size $n$, with $k \ll n$, to ensure data lies in the extreme region.
• Non-asymptotic upper bounds on the expected $L^\infty$ deviation of the empirical stdf from the true stdf are derived using these concentration tools.

Experimental results

Research questions

• RQ1 What non-asymptotic bounds can be established for the convergence rate of the empirical stable tail dependence function in multivariate extreme value analysis?
• RQ2 How can VC-type inequalities be adapted to control maximal deviations in low-probability regions relevant to extreme data?
• RQ3 What is the optimal rate of convergence for empirical stdf estimation under minimal regularity assumptions?
• RQ4 Can finite-sample confidence bounds be derived for the stdf without requiring smoothness or parametric assumptions?
• RQ5 How does the choice of $k$, the number of extreme order statistics, affect the finite-sample performance of the stdf estimator?

Key findings

• The paper establishes a non-asymptotic upper bound on the expected $L^\infty$ deviation of the empirical stdf from the true stdf with rate $O(k^{-1/2})$.
• The concentration inequalities used are specifically designed for low-probability regions and incorporate the hitting probability $p$ of the class, improving finite-sample accuracy.
• The results hold under minimal assumptions—only the existence of the stable tail dependence function is required, without smoothness or parametric restrictions.
• The method enables reliable estimation of tail dependence using only $k$ extreme order statistics, even when $k \ll n$.
• The framework supports finite-sample inference for extreme event classification and risk assessment in applications such as finance, insurance, and environmental modeling.
• The approach provides a theoretical foundation for empirical risk management in multivariate extremes, filling a gap left by purely asymptotic results.