Composite Robust Estimators for Linear Mixed Models

Abstract

The Classical Tukey-Huber Contamination Model (CCM) is a usual framework to describe the mechanism of outliers generation in robust statistics. In a data set with $n$ observations and $p$ variables, under the CCM, an outlier is a unit, even if only one or few values are corrupted. Classical robust procedures were designed to cope with this setting and the impact of observations were limited whenever necessary. Recently, a different mechanism of outliers generation, namely Independent Contamination Model (ICM), was introduced. In this new setting each cell of the data matrix might be corrupted or not with a probability independent on the status of the other cells. ICM poses new challenge to robust statistics since the percentage of contaminated rows dramatically increase with $p$, often reaching more than $50\%$. When this situation appears, classical affine equivariant robust procedures do not work since their breakdown point is $50\%$. For this contamination model we propose a new type of robust methods namely composite robust procedures which are inspired on the idea of composite likelihood, where low dimension likelihood, very often the likelihood of pairs, are aggregate together in order to obtain an approximation of the full likelihood which is more tractable. Our composite robust procedures are build over pairs of observations in order to gain robustness in the independent contamination model. We propose composite S and $\tau$-estimators for linear mixed models. Composite $\tau$-estimators are proved to have an high breakdown point both in the CCM and ICM. A Monte Carlo study shows that our estimators compare favorably with respect to classical S-estimators under the CCM and outperform them under the ICM. One example based on a real data set illustrates the new robust procedure.