Bayesian robust quantile regression and risk measures

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. Besides having the τ-level quantile as a parameter, the SEP distribution has an additional key parameter governing the decay of the tails, making it attractive for robust modeling. Linear and Additive Models (AM) with penalized spline are used to show the exibility 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. To implement the Bayesian inference we propose a new adaptive Metropolis within Gibbs algorithm. Empirical evidence of the statistical properties of the proposed SEP Bayesian quantile regression method is provided through several examples based on both simulated and real datasets.