Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).