Robust modeling using non-elliptically contoured multivariate tt distributions

Models based on multivariate tt distributions are widely applied to analyze data with heavy tails. However, all the marginal distributions of the multivariate tt distributions are restricted to have the same degrees of freedom, making these models unable to describe different marginal heavy-tailedness. We generalize the traditional multivariate tt distributions to non-elliptically contoured multivariate tt distributions, allowing for different marginal degrees of freedom. We apply the non-elliptically contoured multivariate tt distributions to three widely-used models: the Heckman selection model with different degrees of freedom for selection and outcome equations, the multivariate Robit model with different degrees of freedom for marginal responses, and the linear mixed-effects model with different degrees of freedom for random effects and within-subject errors. Based on the normal mixture representation of our tt distribution, we propose efficient Bayesian inferential procedures for the model parameters based on data augmentation and parameter expansion. We show via simulation studies and real data examples that the conclusions are sensitive to the existence of different marginal heavy-tailedness.