A1724
Title: Concentration inequalities for location and scatter halfspace median under contaminated alpha-symmetric distributions
Authors: Filip Bocinec - Charles University (Czech Republic) [presenting]
Stanislav Nagy - Charles University (Czech Republic)
Abstract: In a landmark result, Chen, Gao and Ren (2018) demonstrated that, under the classical Huber contamination model for multivariate elliptically symmetric distributions, the multivariate medians induced by halfspace depth achieve the minimax optimal convergence rate. This result applies both to location and scatter estimation. We extend some of those findings to the broader family of alpha-symmetric distributions, which includes both elliptically symmetric (alpha = 2) and multivariate heavy-tailed (alpha < 2) distributions. In the location estimation scenario, we establish an upper bound on the estimation deviation for the location halfspace median of alpha-symmetric distributions in the Huber contamination model. An analogous result for the standard scatter halfspace median matrix is, however, feasible only under the assumption of elliptical symmetry (alpha = 2). That limitation arises because ellipticity is deeply embedded in the definition of scatter halfspace depth. Therefore, we modify the scatter halfspace depth to better accommodate alpha-symmetric distributions and derive an upper bound for the corresponding alpha-scatter median matrix. Notably, our results hold without any moment assumptions on the underlying distribution, as is common in the literature. Additionally, we identify several key properties of scatter halfspace depth for alpha-symmetric distributions.
