Discriminating Tail Behavior Using Halfspace Depths: Population and Empirical Perspectives

2025-06-01

AI Summary

This paper investigates the convergence rate of the halfspace depth empirical estimator, establishing for the first time a quantitative relationship between this rate and the tail index of the underlying distribution (e.g., Weibull- or Pareto-type tails). Methodologically, it integrates weighted empirical process theory, extreme value analysis, and multivariate nonparametric statistics to derive universal upper bounds on the convergence rate, explicitly characterizing the joint influence of sample size and tail parameters. A key contribution is a novel, depth-based framework for tail-type discrimination—applicable uniformly to both light- and heavy-tailed multivariate distributions—leveraging the asymptotic decay rate of depth values. Extensive simulations and real-data experiments demonstrate that the proposed method substantially improves accuracy in multivariate tail identification, offering a new tool for modeling tail behavior in high-dimensional distributions.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

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.

Problem

• Connecting convergence rates to tail behavior using halfspace depths
• Distinguishing light and heavy tails in multivariate distributions
• Proposing methodology to assess tail behavior from empirical data

Innovation

• Uses halfspace depths for tail behavior analysis
• Employs weighted empirical processes for proofs
• Distinguishes light and heavy tails effectively