Location and scatter halfspace median under α-symmetric distributions

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.

💡 Research Summary

The paper revisits the robust multivariate location and scatter estimation problem based on halfspace depth (HD) and scatter halfspace depth (sHD), extending the seminal results of Chen et al. (2018) beyond the restrictive elliptical (α = 2) setting to the much broader class of α‑symmetric distributions. An α‑symmetric distribution is defined through a characteristic function of the form ψ_X(t)=φ(‖t‖_α), where ‖·‖_α denotes the α‑norm (α>0). This family includes spherical (α=2) distributions, multivariate stable laws (0<α<2), and other heavy‑tailed models, making it highly relevant for modern data‑analytic contexts where outliers and heavy tails are common.

The authors first recall the definition of halfspace depth: D(x;P)=inf_{u∈S^{d−1}} P(⟨X,u⟩≤⟨x,u⟩). For α‑symmetric P, they derive a closed‑form expression D(x;P)=F(−‖x‖_β), where β is the Hölder conjugate of α (β=α/(α−1) for α>1, β=∞ otherwise) and F is the marginal CDF of the first coordinate. This explicit formula shows that the halfspace median is always the origin (μ_hs=0) and that the maximal depth equals ½, regardless of the specific α‑symmetric law.

Next, the paper studies the classical Huber ε‑contamination model, where the observed distribution is (1−ε)P+εQ. By exploiting the monotonicity and Lipschitz continuity of F, together with Hoeffding‑type concentration inequalities, the authors obtain a non‑asymptotic bound for the location estimator μ̂_n=μ_hs(𝑃_n):

 P(‖μ̂_n−μ‖_2 ≤ C·(ε + √