Finite sample breakdown point of Tukey's halfspace median

TL;DR

This paper precisely determines the finite sample breakdown point of Tukey's halfspace median, providing insights into its robustness and the influence of data dimension, complementing existing asymptotic results.

Contribution

It offers the first exact finite sample breakdown point calculation for Tukey's halfspace median, enhancing understanding of its robustness in practical scenarios.

Findings

Finite sample breakdown point of Tukey's median is explicitly characterized.

Dimension significantly affects the robustness of the median.

Results complement existing asymptotic robustness findings.

Abstract

Tukey'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 \emph{finite sample} practical scenario, revealing the dimension effect on the breakdown point robustness and complimenting the existing \emph{asymptotic} breakdown point result.