  Models based on multivariate t distributions are widely applied to analyze
data with heavy tails. However, all the marginal distributions of the
multivariate t 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 t distributions to
non-elliptically contoured multivariate t distributions, allowing for different
marginal degrees of freedom. We apply the non-elliptically contoured
multivariate t 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 t 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 examples that the
conclusions are sensitive to the existence of different marginal
heavy-tailedness.
