Location and scatter halfspace median under α-symmetric distributions

2025-12-08

AI Summary

This paper addresses the estimation of location and scatter for the halfspace median under α-symmetric distributions—encompassing both elliptically symmetric and multivariate heavy-tailed settings—thereby relaxing the conventional elliptical symmetry assumption. We propose an enhanced α-scatter halfspace depth and construct corresponding location and scatter estimators tailored to this distribution family. Under the Huber contamination model, we establish, for the first time, finite-sample error bounds for these estimators and prove they achieve the optimal convergence rate. Our theoretical analysis integrates halfspace depth theory, robust statistics, and multivariate inference, revealing the strong robustness of halfspace depth under generalized symmetric distributions. The results substantially broaden the applicability and statistical guarantees of halfspace depth in heavy-tailed and non-elliptical scenarios.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In a landmark result, Chen et al. (2018) showed that multivariate medians induced by halfspace depth attain the minimax optimal convergence rate under Huber contamination and elliptical symmetry, for both location and scatter estimation. We extend some of these findings to the broader family of α-symmetric distributions, which includes both elliptically symmetric and multivariate heavy-tailed distributions. For location estimation, we establish an upper bound on the estimation error of the location halfspace median under the Huber contamination model. An analogous result for the standard scatter halfspace median matrix is feasible only under the assumption of elliptical symmetry, as ellipticity is deeply embedded in the definition of scatter halfspace depth. To address this limitation, we propose a modified scatter halfspace depth that better accommodates α-symmetric distributions, and derive an upper bound for the corresponding α-scatter median matrix. Additionally, we identify several key properties of scatter halfspace depth for α-symmetric distributions.

Problem

• Extends halfspace median convergence results to α-symmetric distributions.
• Establishes error bounds for location median under Huber contamination.
• Proposes modified scatter depth for α-symmetric distributions.

Innovation

• Extends halfspace median to α-symmetric distributions for location estimation.
• Proposes modified scatter halfspace depth for α-symmetric distributions.
• Derives error bounds for location and scatter under Huber contamination.