Bayesian quantile regression using the skew exponential power distribution

Traditional Bayesian quantile regression relies on the Asymmetric Laplace (AL) distribution due primarily to its satisfactory empirical and theoretical performances. However, the AL displays medium tails and it is not suitable for data characterized by strong deviations from the Gaussian hypothesis. An extension of the AL Bayesian quantile regression framework is proposed to account for fat tails using the Skew Exponential Power (SEP) distribution. Linear and Additive Models (AM) with penalized splines are considered to show the flexibility of the SEP in the Bayesian quantile regression context. Lasso priors are used in both cases to account for the problem of shrinking parameters when the parameters space becomes wide while Bayesian inference is implemented using a new adaptive Metropolis within Gibbs algorithm. Empirical evidence of the statistical properties of the proposed models is provided through several examples based on both simulated and real datasets.