On $\sqrt{n}-$consistency for Bayesian quantile regression based on the misspecified asymmetric Laplace likelihood

The asymmetric Laplace density (ALD) is used as a working likelihood for Bayesian quantile regression. Sriram et al. (2013) derived posterior consistency for Bayesian linear quantile regression based on the misspecified ALD. While their paper also argued for $\sqrt{n}-$consistency, Sriram and Ramamoorthi (2017) highlighted that the argument was only valid for $n^{\alpha}$ rate for $\alpha<1/2$. However, $\sqrt{n}-$rate is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we give sufficient conditions for $\sqrt{n}-$consistency in the more general setting of Bayesian non-linear quantile regression based on ALD. In particular, we derive $\sqrt{n}-$consistency for the Bayesian linear quantile regression. Our approach also enables an interesting extension of the linear case when number of parameters $p$ increases with $n$, where we obtain posterior consistency at the rate $n^{\alpha}$ for $\alpha<1/2$.