Parameter estimations in linear mixed models with heavy tailed distributions

Linear Mixed Models (LMMs) gain its popularity being a comprehensive technique to explain within and between observations variability even for unbalanced data. It's used for clustered, panel, longitudinal data and random blocks in experimental designs with generalized linear and nonlinear forms in addition to linear forms. The main assumption of classical LMM is having normally distributed random effects and error terms. However, there are several situations for that we need to use heavier tails distributions than the (multivariate) normal to handle outliers and/or heavy tailness in data. In this study, we focus on LMM using the multivariate Laplace distribution which is known as the heavy tailed alternative to the normal distribution. Being able to write Laplace distribution as a scale mixture of normal distribution allows us to define the proposed model as hierarchical form of the proposed model. Additionally, the number of parameters of Laplace distribution is less than t-distribution that makes the estimation procedures simpler than the estimations based on the latter one. The parameter estimators of interest are generated with EM algorithm for LMM with Laplace distributed random effects and errror terms and also with only Laplace distributed random effects. A simulation study is provided to illustrate the performance of the Laplace distribution over the normal distribution for LMM. Data sets are generated under the contamination and with outliers. Also, a real data example is used to explore the behavior of the proposed estimators over the counterparts.