Discussion

Abstract

I thank Yang et al. (2015) (henceforth Yang) for a well-written paper that explores the asymptotic validity of the asymmetric Laplace (AL) distribution in Bayesian quantile regression. Asymptotically, the mode of the posterior distribution of the regression parameters is the maximum likelihood estimator of the AL distribution. Yang use this fact to inform a simple correction to the estimate of the posterior variance that dramatically improves coverage probabilities. Yang convincingly present asymptotic and numerical arguments illustrating the improvement.The proposed approach consists of first employing standard frequentist quantile regression of the median of the response on the predictors to estimate the variance of the errors. Then, the parameters are estimated using a Bayesian model with the asymmetric Laplace working likelihood that incorporates a point mass prior for the error variance. Finally, the covariance of the final estimates is used as the 'meat' in a 'sandwich' estimator that is designed to mimic the asymptotic sampling distribution of the regression parameters. The approach borrows asymptotic validity and flexibility from the frequentist domain and the ease of approximating covariance matrices from the Bayesian domain to greatly improve coverage probabilities.The simulation study in the paper introducing the AL in Bayesian quantile regression (Yu & Moyeed, 2001) did not include predictors and assumed standard normal errors. The improvement shown in coverage probabilities in a more challenging simulation setting is quite an achievement. I tested the improvement in another simulation setting, with the assistance of the code from Yang. Define the functions a. / D sign.0:5 / log.