  Many real-life data sets can be analyzed using Linear Mixed Models (LMMs).
Since these are ordinarily based on normality assumptions, under small
deviations from the model the inference can be highly unstable when the
associated parameters are estimated by classical methods. On the other hand,
the density power divergence (DPD) family, which measures the discrepancy
between two probability density functions, has been successfully used to build
robust estimators with high stability associated with minimal loss in
efficiency. Here, we develop the minimum DPD estimator (MDPDE) for independent
but non identically distributed observations in LMMs. We prove the theoretical
properties, including consistency and asymptotic normality. The influence
function and sensitivity measures are studied to explore the robustness
properties. As a data based choice of the MDPDE tuning parameter $\alpha$ is
very important, we propose two candidates as "optimal" choices, where
optimality is in the sense of choosing the strongest downweighting that is
necessary for the particular data set. We conduct a simulation study comparing
the proposed MDPDE, for different values of $\alpha$, with the S-estimators,
M-estimators and the classical maximum likelihood estimator, considering
different levels of contamination. Finally, we illustrate the performance of
our proposal on a real-data example.
