Composite Robust Estimators for Linear Mixed Models

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.