On (in)consistency of M-estimators under contamination

TL;DR

This paper investigates the robustness of various M-estimators for location and scale under contamination, revealing that some popular estimators are inconsistent in asymmetric contamination scenarios, while others remain consistent.

Contribution

It provides a theoretical analysis of the inconsistency of common robust estimators under contamination and identifies conditions for the consistency of the Tukey estimator.

Findings

Median and Huber estimators are inconsistent under asymmetric contamination.

Tukey estimator remains consistent under contamination.

Standard robust scale estimators like IQR and MAD are inconsistent under contamination.

Abstract

We consider robust location-scale estimators under contamination. We show that commonly used robust estimators such as the median and the Huber estimator are inconsistent under asymmetric contamination, while the Tukey estimator is consistent. In order to make nuisance parameter free inference based on the Tukey estimator a consistent scale estimator is required. However, standard robust scale estimators such as the interquartile range and the median absolute deviation are inconsistent under contamination.