Composite Robust Estimators for Linear Mixed Models

ABSTRACT

The classical Tukey–Huber contamination model (CCM) is a commonly adopted framework to describe the mechanism of outliers generation in robust statistics. Given a dataset with n observations and p variables, under the CCM, an outlier is a unit, even if only one or a few values are corrupted. Classical robust procedures were designed to cope with this type of outliers. Recently, a new mechanism of outlier generation was introduced, namely, the independent contamination model (ICM), where the occurrences that each cell of the data matrix is an outlier are independent events and have the same probability. ICM poses new challenges to robust statistics since the percentage of contaminated rows dramatically increase with p, often reaching more than 50% whereas classical affine equivariant robust procedures have a breakdown point of 50% at most. For ICM, we propose a new type of robust methods, namely, composite robust procedures that are inspired by the idea of composite likelihood, where low-dimension likelihood, very often the likelihood of pairs, are aggregated to obtain a tractable approximation of the full likelihood. Our composite robust procedures are built on pairs of observations to gain robustness in the ICM. We propose composite τ-estimators for linear mixed models. Composite τ-estimators are proved to have a high breakdown point both in the CCM and ICM. A Monte Carlo study shows that while classical S-estimators can only cope with outliers generated by the CCM, the estimators proposed here are resistant to both CCM and ICM outliers. Supplementary materials for this article are available online.

KEYWORDS:

• Composite τ-estimators
• Independent contamination model
• Robust estimation
• Tukey–Huber contamination model

Supplementary Material

Supplementary Material with the derivation of the estimating equations, discussion on computational aspects and algorithms, proofs of theorems and R code for the example, and the Monte Carlo experiment is available online.

An R package robustvarComp is available in the Comprehensive R Archive Network at cran.r-project.org/web/packages/robustvarComp/index.html. The package implements composite S-estimators and τ-estimators and the CVFS estimator for linear mixed models.

Funding

This research was partially supported by the Italian-Argentinian project “Metodi robusti per la previsione del costo e della durata della degenza ospedaliera” funded inside the collaboration program of MINCYT-MAE AR14MO6. Víctor Yohai research was also partially supported by Grants 20020130100279 from Universidad of Buenos Aires, PIP 112-2008-01-00216 and 112-2011-01-00339 from CONICET and PICT 2011-0397 from ANPCYT. All statistical analyses were performed on SCSCF (www.dais.unive.it/scscf), a multiprocessor cluster system owned by Ca’ Foscari University of Venice running under GNU/Linux.