Empirical tail dependence functions in high dimensions: uniform linearizations and inference

TL;DR

This paper develops new theoretical tools for inference on high-dimensional tail dependence, providing uniform linearizations, error bounds, and bootstrap validity for empirical tail dependence estimators.

Contribution

It introduces foundational theory for inference on empirical tail dependence functions in high dimensions, including uniform linearization and bootstrap methods.

Findings

Finite-sample probability bounds for rank-based estimators

High-dimensional CLTs and bootstrap validity established

Applications to tail dependence parameter estimation and spatial isotropy

Abstract

The analysis of extremal dependence in high dimensions has recently attracted considerable interest. Existing methodology primarily focuses on modeling and estimation of extremal dependence structures, often supported by concentration bounds for empirical tail quantities. However, comparatively little is known about general inferential procedures in high-dimensional extremes. In this paper, we develop foundational theory enabling inference for methods based on empirical tail dependence coefficients and stable tail dependence functions. These estimators are constructed from ranks, which complicates distributional approximations since the stochastic fluctuations of the ranks interfere with those arising from the unknown tail dependence. We establish uniform linearization results for empirical stable tail dependence functions in the form of finite-sample probability bounds that quantify the error of the rank linearization uniformly over collections of coordinates. Within an asymptotic framework, these bounds allow the dimension to grow exponentially with the effective sample size while preserving the validity of the linear approximation. Moreover, we derive high-dimensional central limit theorems and establish the validity of multiplier bootstrap procedures for collections of empirical tail dependence statistics. We illustrate the usefulness of the results through two applications: uniform expansions for M-estimators of tail dependence parameters and inference for spatial isotropy based on collections of tail dependence functions..