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On (in)consistency of M-estimators under contamination

Jens Klooster, Bent Nielsen

TL;DR

This paper studies the robustness of M-estimators for location and scale under fixed contamination, showing that non-redescending estimators like the median and Huber can be inconsistent under asymmetric contamination while redescending estimators like Tukey can be consistent when the scale is estimated consistently. It develops an asymptotic framework under a contamination model, deriving boundedness results, an oracle property for redescending estimators, and explicit inconsistency results for non-redescending ones; it also demonstrates that standard robust scale estimators (IQR, MAD) are themselves inconsistent under contamination. The practical implication is that valid inference under contamination requires modelling the contamination and/or using alternatives such as Least Trimmed Squares (LTS), which exhibits an oracle property and nuisance-parameter-free inference under the contamination model. The simulations corroborate that Tukey with a consistent scale can achieve consistency, but LTS often provides superior finite-sample performance across contaminated scenarios, underscoring the value of robust, contamination-aware approaches in location-scale problems.

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.

On (in)consistency of M-estimators under contamination

TL;DR

This paper studies the robustness of M-estimators for location and scale under fixed contamination, showing that non-redescending estimators like the median and Huber can be inconsistent under asymmetric contamination while redescending estimators like Tukey can be consistent when the scale is estimated consistently. It develops an asymptotic framework under a contamination model, deriving boundedness results, an oracle property for redescending estimators, and explicit inconsistency results for non-redescending ones; it also demonstrates that standard robust scale estimators (IQR, MAD) are themselves inconsistent under contamination. The practical implication is that valid inference under contamination requires modelling the contamination and/or using alternatives such as Least Trimmed Squares (LTS), which exhibits an oracle property and nuisance-parameter-free inference under the contamination model. The simulations corroborate that Tukey with a consistent scale can achieve consistency, but LTS often provides superior finite-sample performance across contaminated scenarios, underscoring the value of robust, contamination-aware approaches in location-scale problems.

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.

Paper Structure

This paper contains 22 sections, 6 theorems, 27 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. M-estimators for location with known scale
  4. M-estimators with unknown scale
  5. The Huber contamination model
  6. The infinitesimal approach
  7. The breakdown point
  8. Some drawbacks of standard practice
  9. A different approach: modelling contamination
  10. Asymptotic results
  11. Asymptotic sequence of models
  12. Boundedness
  13. Asymptotics for redescending M-estimators
  14. Inconsistency of non-redescending M-estimators
  15. Scale estimation
  16. ...and 7 more sections

Key Result

Theorem 3.1

Suppose Assumptions assumption-rho, assumption-boundedness and that $(i)$: $\psi_* > 0$ and $\lambda > 1/2$; or $(ii)$: $\psi_* = 0$ and $\lambda > 1/(2 - \tilde{\rho}_{\varsigma}/\rho_*)$. Then, for large $n$, sets with large probability exist on which all minimizers $\hat{\mu}$ of $R_n(\mu)$ are u

Figures (1)

  • Figure 1: Bias of different M-estimators in relation to the consistency factor $\varsigma$. The red vertical line indicates the smallest $\varsigma$ for which boundedness holds for Tukey's estimator according to Theorem \ref{['theorem-bounded']}$(ii)$.

Theorems & Definitions (14)

  • Theorem 3.1: Boundedness
  • Theorem 3.2: Closeness
  • Theorem 3.3: Asymptotic distribution
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • proof : Proof of Theorem \ref{['theorem-bounded']}
  • proof : Proof of Theorem \ref{['theorem-consistency']}
  • proof : Proof of Theorem \ref{['theorem-distribution']}
  • proof : Proof of Theorem \ref{['theorem-non-consistency']}
  • ...and 4 more