Some results on the computing of Tukey's halfspace medain

TL;DR

This paper investigates the depth of Tukey's median for empirical distributions, providing a sharper upper bound that is practical for large datasets and reduces computational complexity.

Contribution

It introduces a new, lower upper bound for Tukey's median depth applicable to general position data and fixed sample sizes, improving computational efficiency.

Findings

New upper bound for Tukey median depth in empirical distributions

Bound is lower than previous literature, more practical for fixed sample sizes

Results help reduce computational burden for high-dimensional data

Abstract

Depth of the Tukey median is investigated for empirical distributions. A sharper upper bound is provided for this value for data sets in general position. This bound is lower than the existing one in the literature, and more importantly derived under the \emph{fixed} sample size practical scenario. Several results obtained in this paper are interesting theoretically and useful as well to reduce the computational burden of the Tukey median practically when $p$ is large relative to large $n$.