Polar Depth for Potentially Heavy-Tailed Data

2026-05-29

AI Summary

This work addresses the challenge of measuring extremal behavior and detecting anomalies in heavy-tailed multivariate distributions by introducing a novel statistical depth function—Polar Depth—that uniquely integrates statistical depth with the theory of regular variation in polar coordinates. By representing data in polar form, the method captures the centrality of extreme observations whose support lies in half-spaces, offering a natural characterization of tail dependence structures. Theoretically, it is shown that as the threshold tends to infinity, the proposed depth converges to the depth induced by the limiting angular measure of the underlying regularly varying distribution. Empirical evaluations demonstrate that Polar Depth substantially outperforms conventional halfspace depth in both ranking extremes and identifying anomalies, thereby providing a more principled and effective framework for extremal analysis of heavy-tailed data.

Abstract

Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as Polar Depth. The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a regularly varying random variable, beyond asymptotically large thresholds, once its marginals have been appropriately normalized. Not only does the polar depth function make it easy to order the extreme values taken by a heavy-tailed random variable X and finds natural applications in anomaly detection, but it is also possible to show, as we prove it under appropriate assumptions in this article, that the polar depth of the largest observations, i.e. observations X which norm is larger than t>0, converges to the polar depth of the limiting distribution as t converges to infinity. Although designed to quantify the depth of multivariate extremes, the polar depth is interesting in its own right, insofar as this notion is more relevant for distributions whose support is included in a halfspace than the alternatives proposed in the literature, the halfspace depth in particular. Here, we demonstrate its properties and analyze statistical issues related to its estimation from both finite-sample and asymptotic points of view. We present numerical results to empirically demonstrate its relevance, particularly for the statistical analysis of extreme observations and more specifically for the identification of anomalies among them.

Problem

• statistical depth
• heavy-tailed data
• extreme value analysis
• anomaly detection
• multivariate extremes

Innovation

• Polar Depth
• Heavy-tailed distributions
• Extreme value analysis
• Statistical depth
• Anomaly detection