Bayesian Quantile Regression using the Skew Exponential Power Distribution

Traditional Bayesian quantile regression relies on the Asymmetric Laplace Distribution (ALD) due primarily to its satisfactory empirical and theoretical performances. However, the ALD displays medium tails and is not suitable for data characterized by strong deviations from the Gaussian hypothesis. In this paper, we propose an extension of the ALD Bayesian quantile regression framework to account for fat tails using the Skew Exponential Power (SEP) distribution. Linear and Additive Models (AM) with penalized spline are used to show the flexibility of the SEP in the Bayesian quantile regression context. Lasso priors are used to account for the problem of shrinking parameters when the parameters space becomes wide. We propose an adaptive Metropolis–Hastings algorithm in the linear model, and an adaptive Metropolis within Gibbs in the AM framework. Empirical evidence of the statistical properties of the model is provided through several examples based on both simulated and real datasets.