Posterior rates of convergence for composite quantile regression

Composite quantile regression is based on the convex combination of single quantile quantile loss functions and enjoys many advantages over single quantile regression. The Bayesian extension is based on the finite mixture of asymmetric Laplace densities. This article mainly aims to contribute to the theoretical justification of Bayesian composite quantile regression from the perspective of Bayesian density estimation. As such, we further show that the asymmetric Laplace distribution can be used for Bayesian density estimation in general. We obtain upper bounds on rates of convergence for mixtures of asymmetric Laplace densities. For finite mixtures, we obtain the parametric rate up to a logarithmic factor, and a slower rate for infinite mixture.