Fast computation of Tukey trimmed regions and median in dimension $p>2$

TL;DR

This paper introduces two efficient algorithms for computing Tukey trimmed regions and median in higher dimensions, significantly improving computational speed and providing theoretical bounds, with extensive simulation validation.

Contribution

It presents a novel, faster algorithm for Tukey region computation in dimensions greater than two, along with bounds on facets and an extension to the Tukey median.

Findings

The fast algorithm outperforms the naive one in speed while maintaining accuracy.

A strict bound on the number of facets of Tukey regions is established.

Simulation results confirm the efficiency and correctness of the proposed methods.

Abstract

Given data in $\mathbb{R}^{p}$, a Tukey $\kappa$-trimmed region is the set of all points that have at least Tukey depth $\kappa$ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension $p > 2$. We construct two novel algorithms to compute a Tukey $\kappa$-trimmed region, a na\"{i}ve one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the na\"{i}ve one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median.