Abstract: Abstract: Quantile regression is a powerful data analysis tool that accommodates heterogeneous covariate-response relationships. We find that by coupling the asymmetric Laplace working likelihood with appropriate shrinkage priors, we can deliver pseudo-Bayesian inference that adapts automatically to possible sparsity in quantile regression analysis. After a suitable adjustment on the posterior variance, the proposed method provides asymptotically valid inference under heterogeneity. Furthermore, the proposed approach leads to oracle asymptotic efficiency for the active (nonzero) quantile regression coefficients, and super-efficiency for the non-active ones. By avoiding dichotomous variable selection, the Bayesian computational framework demonstrates desirable inference stability with respect to tuning parameter selection. Our work helps to uncloak the value of Bayesian computational methods in frequentist inference for quantile regression.

Key words and phrases: Asymmetric Laplace distribution, increasing dimension, optimal weighting, posterior asymptotics, shrinkage prior, working likelihood.