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.