Year: 2025
Type: article
Abstract: In a landmark result, Chen et al. [(2018), ‘Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model’, The Annals of Statistics, 46(5), 1932–1960] showed that multivariate medians induced by halfspace depth attain the minimax optimal convergence rate under Huber contamination and ellipticity, for both location and scatter. We extend some of these findings to the broader family of α-symmetric distributions, including elliptical and heavy-tailed distributions. For location, we establish an upper bound on the estimation error of the location halfspace median under Huber contamination. An analogous result for the scatter halfspace median is feasible only under ellipticity, which is deeply embedded in the definition of scatter halfspace depth. To overcome this, we introduce the α-scatter halfspace depth, specifically tailored for α-symmetric distributions. The corresponding α-scatter median provides a novel robust estimator of multivariate scatter under heavy-tailed and contaminated settings. We study its fundamental properties, including consistency and invariance, and derive the convergence rate. Additionally, we identify several properties of the usual scatter halfspace depth.
Authors: Filip Bočinec, Stanislav Nagy
Institution: Charles University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417436781