Multivariate linear regression with non-normal errors: a proposal based on mixture models

Multivariate regression analysis is a well-known technique used to predict values of d responses from a set of p regressors. Usually, it is assumed that the error term has a multivariate normal distribution with a zero mean vector and a positive definited covariance matrix. However, in many real situations this assumption may be unrealistic. This problem has been addressed by several authors (see for example Ferreira and Steel (2004), Batsidis and Zografos (2008) and the references therein), through the introduction of multivariate skewed and/or heavy-tailed distributions. In this paper, we approach this problem in a semi-parametric setting, by modeling the error term distribution through a finite mixture of d-dimensional Gaussian components. This solution generalizes the proposal described by Bartolucci and Scaccia (2005) in the context of univariate regression analysis (d=1). The parameters of this mixture model and the regression coefficients are estimated by the EM algorithm. The choice of the number of mixture components is performed through model selection criteria. The performances of this new method and some of the existing ones are compared through a Monte Carlo experiment. In particular, this comparison focuses on the bias and mean-square error of the regression coefficient estimators. In conclusion, some real-data examples are presented.