Nonparametric quantile inference with Dirichlet processes

This article is concerned with nonparametric inference for quantiles from a Bayesian perspective, using the Dirichlet process. The posterior distribution for quantiles is characterized, enabling also explicit formulae for posterior mean and variance. Unlike the Bayes estimator for the distribution function, our Bayes estimator for the quantile function is a smooth curve. A Bernstein-von Mises type theorem is given, exhibiting the limiting posterior distribution of the quantile process. Links to kernel smoothed quantile estimators are provided. As a side product we develop an automatic nonparametric density estimator, free of smoothing parameters, with support exactly matching that of the data range. Nonparametric Bayes estimators are also provided for other quantile-related quantities, including the Lorenz curve and the Gini index, for Doksum's shift curve and for Parzen's comparison distribution in two-sample situations, and finally for the quantile regression function in situations with covariates.