Robust Bayesian model averaging for linear regression models with heavy-tailed errors

Our goal is to develop a Bayesian model averaging technique in linear regression models that accommodates heavier tailed error densities than the normal distribution. Motivated by the use of the Huber loss function in the presence of outliers, the Bayesian Huberized lasso with hyperbolic errors has been proposed and recently implemented in the literature. Since the Huberized lasso cannot enforce regression coefficients to be exactly zero, we propose a Bayesian variable selection approach with spike and slab priors to address sparsity more effectively. The shapes of the hyperbolic and the Student-<i>t</i> density functions differ. Furthermore, the tails of a hyperbolic distribution are less heavy compared to those of a Cauchy distribution. Thus, we propose a flexible regression model with an error distribution encompassing both the hyperbolic and the Student-<i>t</i> family of distributions, with an unknown tail heaviness parameter, that is estimated based on the data. It is known that the limiting form of both the hyperbolic and the Student-<i>t</i> distributions is a normal distribution. We develop an efficient Gibbs sampler for posterior computation. Through simulation studies and analyzes of real datasets, we show that our method is competitive with various state-of-the-art methods.