On Bayesian robust regression with diverging number of predictors

This paper concerns the robust regression model when the number of predictors and the number of observations grow in a similar rate. Theory for M-estimators in this regime has been recently developed by several authors (El Karoui et al., 2013; Bean et al., 2013; Donoho and Montanari, 2013). Motivated by the inability of M-estimators to successfully estimate the Euclidean norm of the coefficient vector, we consider a Bayesian framework for this model. We suggest a two-component mixture of normals prior for the coefficients and develop a Gibbs sampler procedure for sampling from relevant posterior distributions, while utilizing a scale mixture of normal representation for the error distribution. Unlike M-estimators, the proposed Bayes estimator is consistent in the Euclidean norm sense. Simulation results demonstrate the superiority of the Bayes estimator over traditional estimation methods.