Discriminating Tail Behavior Using Halfspace Depths: Population and Empirical Perspectives

TL;DR

This paper investigates the relationship between halfspace depth convergence rates and tail behavior of distributions, providing a methodology to assess tail heaviness in multivariate data with theoretical and practical insights.

Contribution

It establishes a connection between empirical halfspace depths and tail behavior, offering a new approach to distinguish light and heavy tails in multivariate distributions.

Findings

Method to differentiate tail types using halfspace depths

Theoretical link between convergence rates and tail behavior

Application demonstrated on simulated and real data

Abstract

We study the empirical version of halfspace depths with the objective of establishing a connection between the rates of convergence and the tail behaviour of the corresponding underlying distributions. The intricate interplay between the sample size and the parameter driving the tail behaviour forms one of the main results of this analysis. The chosen approach is mainly based on weighted empirical processes indexed by sets by Alexander (1987), which leads to relatively direct and elegant proofs, regardless of the nature of the tail. This method is further enriched by our findings on the population version, which also enable us to distinguish between light and heavy tails. These results lay the foundation for our subsequent analysis of the empirical versions. Building on these theoretical insights, we propose a methodology to assess the tail behaviour of the underlying multivariate distribution of a sample, which we illustrate on simulated data. The study concludes with an application to a real-world dataset.