Posterior consistency of bayesian quantile regression based on the misspecied asymmetric laplace density

Abstract

We explore an asymptotic justication for the widely used and em-pirically veried approach of assuming an asymmetric Laplace distribution (ALD) for the response in Bayesian Quantile Regression. Based on empirical ndings, Yu and Moyeed (2001) argued that the use of ALD is satisfactory even if it is not the true underlying distribution. We provide a justication to this claim by establishing posterior consistency and deriving the rate of convergence under the ALD misspecication. Related literature on misspecied models focuses mostly on i.i.d. models which in the regression context amounts to considering i.i.d. random covariates with i.i.d. errors. We study the behavior of the posterior for the mis-specied ALD model with independent but non identically distributed response in the presence of non-random covariates. Exploiting the specic form of ALD helps us derive conditions that are more intuitive and easily seen to be satised by a wide range of potential true underlying probability distributions for the response. Through simulations, we demonstrate our result and also nd that the robustness of the posterior that holds for ALD fails for a Gaussian formulation, thus providing further support for the use of ALD models in quantile regression. � 2013 International Society for Bayesian Analysis.