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.

翻译：在Chen等人（2018）的里程碑式研究中，他们证明了在半空间深度导出的多元中位数在Huber污染模型与椭圆对称性假设下，对于位置与散度估计均能达到极小极大最优收敛速率。我们将部分结论推广至更广泛的α对称分布族，该族包含椭圆对称分布与多元重尾分布。针对位置估计，我们建立了Huber污染模型下位置半空间中位数估计误差的上界。对于标准散度半空间中位数矩阵的类似结果，仅在椭圆对称性假设下可行，因为椭圆特性深植于散度半空间深度的定义中。为克服此限制，我们提出了一种改进的散度半空间深度，以更好地适应α对称分布，并推导了相应α散度中位数矩阵的上界。此外，我们还识别了α对称分布下散度半空间深度的若干关键性质。