  Traditional Bayesian quantile regression relies on the Asymmetric Laplace
distribution (ALD) mainly because of its satisfactory empirical and theoretical
performances. However, the ALD displays medium tails and it 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. Beside having the $\tau$-level quantile as parameter, the SEP
distribution has an additional key parameter governing the decay of the tails,
making it attractive for robust modeling of conditional quantiles at different
confidence levels. Linear and Generalized Additive Models (GAM) with penalized
spline are considered to show the flexibility of the SEP in the Bayesian
quantile regression context. Lasso priors are considered in both cases to
account for shrinking parameters problem when the parameters space becomes
wide. To implement the Bayesian inference we propose a new adaptive
Metropolis--Hastings algorithm in the linear model and an adaptive Metropolis
within Gibbs one in the GAM framework. Empirical evidence of the statistical
properties of the proposed SEP Bayesian quantile regression method is provided
through several example based on simulated and real dataset.
