A New Perspective on Robust $M$-Estimation: Finite Sample Theory and Applications to Dependence-Adjusted Multiple Testing

TL;DR

This paper develops a finite sample theory for an adaptive Huber estimator that remains robust with heavy-tailed errors, providing new concentration inequalities and applications to multiple testing procedures.

Contribution

It introduces a novel nonasymptotic analysis of the adaptive Huber estimator with divergence of the tuning parameter, extending robustness to heavy-tailed data.

Findings

Derived sub-Gaussian deviation bounds for the estimator

Established nonasymptotic Bahadur representation under finite second moments

Proved asymptotic control of false discovery rate in multiple testing

Abstract

Heavy-tailed errors impair the accuracy of the least squares estimate, which can be spoiled by a single grossly outlying observation. As argued in the seminal work of Peter Huber in 1973 [{\it Ann. Statist.} {\bf 1} (1973) 799--821], robust alternatives to the method of least squares are sorely needed. To achieve robustness against heavy-tailed sampling distributions, we revisit the Huber estimator from a new perspective by letting the tuning parameter involved diverge with the sample size. In this paper, we develop nonasymptotic concentration results for such an adaptive Huber estimator, namely, the Huber estimator with the tuning parameter adapted to sample size, dimension, and the variance of the noise. Specifically, we obtain a sub-Gaussian-type deviation inequality and a nonasymptotic Bahadur representation when noise variables only have finite second moments. The nonasymptotic results further yield two conventional normal approximation results that are of independent interest, the Berry-Esseen inequality and Cram\'er-type moderate deviation. As an important application to large-scale simultaneous inference, we apply these robust normal approximation results to analyze a dependence-adjusted multiple testing procedure for moderately heavy-tailed data. It is shown that the robust dependence-adjusted procedure asymptotically controls the overall false discovery proportion at the nominal level under mild moment conditions. Thorough numerical results on both simulated and real datasets are also provided to back up our theory.