On posterior concentration rates for Bayesian quantile regression based on the misspecified asymmetric Laplace likelihood

TL;DR

This paper investigates the rate at which Bayesian quantile regression with the misspecified asymmetric Laplace likelihood converges, establishing conditions for -consistency and extending results to non-linear models.

Contribution

It derives posterior concentration rates for Bayesian quantile regression with the misspecified ALD, including -consistency conditions and non-linear model examples.

Findings

Established -consistency conditions for Bayesian linear quantile regression.

Extended posterior concentration results to non-linear quantile regression models.

Provided sufficient conditions for convergence rates in misspecified Bayesian models.

Abstract

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 a rate less than $\sqrt{n}$. So, the question of $\sqrt{n}-$rate has remained unaddressed, but is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we derive results to obtain posterior consistency rates for Bayesian quantile regression based on the misspecified ALD. We derive our results in a slightly general setting where the quantile function can possibly be non-linear. In particular, we give sufficient conditions for $\sqrt{n}-$consistency for the Bayesian linear quantile regression using ALD. We also provide examples of Bayesian non-linear quantile regression.