Bayesian analysis of testing general hypotheses in linear models with spherically symmetric errors

Abstract

We consider Bayesian analysis for testing the general linear hypotheses in linear models with spherically symmetric errors. These error distributions not only include some of the classical linear models as special cases, but also reduce the influence of outliers and result in a robust statistical inference. Meanwhile, the design matrix is not necessarily of full rank. By appropriately modifying mixtures of g-priors for the regression coefficients under some general linear constraints, we derive closed-form Bayes factors in terms of the ratio between two Gaussian hypergeometric functions. The proposed Bayes factors rely on the data only through the modified coefficient of determinations of the two models and are shown to be independent of the error distributions, so long as they are spherically symmetric. Moreover, we establish the results of the model selection consistency with the proposed Bayes factors in the model settings with a full-rank design matrix when the number of parameters increases with the sample size. We carry out simulation studies to assess the finite sample performance of the proposed methodology. The presented results extend some existing Bayesian testing procedures in the literature.