The Limit of Finite Sample Breakdown Point of Tukey’s Halfspace Median for General Data

Abstract

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 the literature. In this paper, we establish the result under weaker assumptions imposed on underlying distribution (weak smoothness) and on data set (not necessary in general position). The refined representation of Tukey’s sample depth regions for data set not necessary in general position is also obtained, as a by-product of our derivation.