Improved performance guarantees for Tukey's median

TL;DR

This paper provides improved theoretical performance guarantees for Tukey's median in high-dimensional data, especially under elliptically symmetric distributions, with implications for robust statistics and data analysis.

Contribution

The authors derive tighter bounds on the diameter of empirical Tukey's median set and extend results to affine equivariant estimators, enhancing understanding of robustness in multivariate data.

Findings

Diameter of empirical Tukey's median scales as o(n^{-1/2})

For bivariate normal data, diameter is O(n^{-3/4} log^{1/2}(n)) with high probability

Affine equivariance improves concentration bounds for empirical processes

Abstract

Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with John Tukey, is among the most popular. Tukey's depth has found applications in robust statistics, graph theory, and the study of elections and social choice. We present improved performance guarantees for empirical Tukey's median, a deepest point associated with a given sample, when the data-generating distribution is elliptically symmetric and possibly anisotropic. Some of our results remain valid in the wider class of affine equivariant estimators. As a corollary of our bounds, we show that the typical diameter of the set of all empirical Tukey's medians scales like $o(n^{-1/2})$ where $n$ is the sample size. Moreover, when the data follow the bivariate normal distribution, we prove that with high probability, the diameter is of order $O(n^{-3/4}\log^{1/2}(n))$. On the technical side, we show how affine equivariance can be leveraged to improve concentration bounds; moreover, we develop sharp strong approximation results for empirical processes indexed by halfspaces that could be of independent interest.