Parsimonious mixtures of seemingly unrelated contaminated normal regression models

In recent years, the research into linear multivariate regression based on finite mixture models has been intense. With such an approach, it is possible to perform regression analysis for a multivariate response by taking account of the possible presence of several unknown latent homogeneous groups, each of which is characterised by a different linear regression model. For a continuous multivariate response, mixtures of normal regression models are usually employed. However, in real data, it is not unusual to observe mildly atypical observations that can negatively affect the estimation of the regression parameters under a normal distribution in each mixture component. Furthermore, in some fields of research, a multivariate regression model with a different vector of covariates for each response should be specified, based on some prior information to be conveyed in the analysis. To take account of all these aspects, mixtures of contaminated seemingly unrelated normal regression models have been recently developed. A further extension of such an approach is presented here so as to ensure parsimony, which is obtained by imposing constraints on the group-covariance matrices of the responses. A description of the resulting parsimonious mixtures of seemingly unrelated contaminated regression models is provided together with the results of a numerical study based on the analysis of a real dataset, which illustrates their practical usefulness.