Models based on multivariate t distributions are widely applied to analyzedata with heavy tails. However, all the marginal distributions of themultivariate t distributions are restricted to have the same degrees offreedom, making these models unable to describe different marginalheavy-tailedness. We generalize the traditional multivariate t distributions tonon-elliptically contoured multivariate t distributions, allowing for differentmarginal degrees of freedom. We apply the non-elliptically contouredmultivariate t distributions to three widely-used models: the Heckman selectionmodel with different degrees of freedom for selection and outcome equations,the multivariate Robit model with different degrees of freedom for marginalresponses, and the linear mixed-effects model with different degrees of freedomfor random effects and within-subject errors. Based on the Normal mixturerepresentation of our t distribution, we propose efficient Bayesian inferentialprocedures for the model parameters based on data augmentation and parameterexpansion. We show via simulation studies and real examples that theconclusions are sensitive to the existence of different marginalheavy-tailedness.
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