The limit of finite sample breakdown point of Tukey's halfspace median for general data

TL;DR

This paper extends the understanding of Tukey's halfspace median by establishing its finite sample breakdown point limit under weaker assumptions on data symmetry and position, and provides a new representation of depth regions.

Contribution

It proves the breakdown point limit under less restrictive conditions and derives a new representation of Tukey's depth regions without the general position assumption.

Findings

Breakdown point limit is 1/3 under weaker conditions.

Representation of depth regions is obtained without general position.

Results generalize previous findings to broader data settings.

Abstract

Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey's halfspace median ($\frac{1} {3}$) has been obtained in literature. In this paper, we establish the result under \emph{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 \emph{not necessary in general position} is also obtained, as a by-product of our derivation.