Bayesian non-parametric simultaneous quantile regression for complete and grid data

Bayesian methods for non-parametric quantile regression have been considered with multiple continuous predictors ranging values in the unit interval. Two methods are proposed based on assuming that either the quantile function or the distribution function is smooth in the explanatory variables and is expanded in tensor product of B-spline basis functions. Unlike other existing methods of non-parametric quantile regressions, the proposed methods estimate the whole quantile function instead of estimating on a grid of quantiles. Priors on the coefficients of the B-spline expansion are put in such a way that the monotonicity of the estimated quantile levels are maintained unlike local polynomial quantile regression methods. The proposed methods are also modified for quantile grid data where only the percentile range of each response observations are known. A comparative simulation study of the performances of the proposed methods and some other existing methods are provided in terms of prediction mean squared errors and mean L1-errors over the quartiles. The proposed methods are used to estimate the quantiles of US household income data and North Atlantic hurricane intensity data.