Non-asymptotic analysis and inference for an outlyingness induced winsorized mean

TL;DR

This paper introduces an outlyingness induced winsorized mean estimator that achieves optimal robustness against 50% contamination, maintains sub-Gaussian performance, and is computationally efficient in finite samples.

Contribution

It proposes a novel winsorized mean estimator with maximal robustness and sub-Gaussian performance, improving robustness and efficiency over existing estimators.

Findings

Resists up to 50% data contamination without breakdown.

Achieves sub-Gaussian performance in uncontaminated samples.

Computable in linear time with bounded error in finite samples.

Abstract

Robust estimation of a mean vector, a topic regarded as obsolete in the traditional robust statistics community, has recently surged in machine learning literature in the last decade. The latest focus is on the sub-Gaussian performance and computability of the estimators in a non-asymptotic setting. Numerous traditional robust estimators are computationally intractable, which partly contributes to the renewal of the interest in the robust mean estimation. Robust centrality estimators, however, include the trimmed mean and the sample median. The latter has the best robustness but suffers a low-efficiency drawback. Trimmed mean and median of means, %as robust alternatives to the sample mean, and achieving sub-Gaussian performance have been proposed and studied in the literature. This article investigates the robustness of leading sub-Gaussian estimators of mean and reveals that none of them can resist greater than $25\%$ contamination in data and consequently introduces an outlyingness induced winsorized mean which has the best possible robustness (can resist up to $50\%$ contamination without breakdown) meanwhile achieving high efficiency. Furthermore, it has a sub-Gaussian performance for uncontaminated samples and a bounded estimation error for contaminated samples at a given confidence level in a finite sample setting. It can be computed in linear time.