SummaryTukey's halfspace median (HM), servicing as the multivariate counterpart of the univariate median, has been introduced and extensively studied in the literature. It is supposed and expected to preserve robustness property (the most outstanding property) of the univariate median. One of prevalent quantitative assessments of robustness is finite sample breakdown point (FSBP). Indeed, the FSBP of many multivariate medians have been identified, except for the most prevailing one-the Tukey's halfspace median. This paper presents a precise result on FSBP for Tukey's halfspace median. The result here depicts the complete prospect of the global robustness of HM in the finite sample practical scenario, revealing the dimension effect on the breakdown point robustness and complimenting the existing asymptotic breakdown point result.

Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey's halfspace median ( 1 3 ) has been obtained in literature. In this paper, we establish the result under weaker assumption imposed on underlying distribution (halfspace symmetry) and on data set (not necessary in general position). The representation of Tukey's sample depth regions for data set not necessary in general position is also obtained, as a byproduct of our derivation.