Improved Rademacher symmetrization through a Wasserstein based measure of asymmetry

TL;DR

This paper introduces a Wasserstein-based measure to improve symmetrization inequalities, providing tighter bounds and diverse applications in statistical testing, confidence set construction, and variance bounding.

Contribution

It presents a novel Wasserstein-based asymmetry measure and an empirical bootstrap bound, enhancing symmetrization techniques for various statistical applications.

Findings

Tighter empirical bounds on symmetrization corrections.

Effective testing for data symmetry.

Improved constants in inequalities for Banach space variables.

Abstract

We propose of an improved version of the ubiquitous symmetrization inequality making use of the Wasserstein distance between a measure and its reflection in order to quantify the symmetry of the given measure. An empirical bound on this asymmetric correction term is derived through a bootstrap procedure and shown to give tighter results in practical settings than the original uncorrected inequality. Lastly, a wide range of applications are detailed including testing for data symmetry, constructing nonasymptotic high dimensional confidence sets, bounding the variance of an empirical process, and improving constants in Nemirovski style inequalities for Banach space valued random variables.