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We consider a Bayesian method for simultaneous quantile regression on a real variable. By monotone transformation, we can make both the response variable and the predictor variable take values in the unit interval. A representation of quantile function is given by a convex combination of two monotone increasing functions ξ1 and ξ2 not depending on the prediction variables. In a Bayesian approach, a prior is put on quantile functions by putting prior distributions on ξ1 and ξ2. The monotonicity constraint on the curves ξ1 and ξ2 are obtained through a spline basis expansion with coefficients increasing and lying in the unit interval. We put a Dirichlet prior distribution on the spacings of the coefficient vector. A finite random series based on splines obeys the shape restrictions. We compare our approach with a Bayesian method using Gaussian process prior through an extensive simulation study and some other Bayesian approaches proposed in the literature. An application to a data on hurricane activities in the Atlantic region is given. We also apply our method on region-wise population data of USA for the period 1985-2010.

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In this paper, we consider Bayesian methods for non-parametric quantile regressions with multiple continuous predictors ranging values in the unit interval. In the first method, the quantile function is assumed to be smooth over the explanatory variable and is expanded in tensor product of B-spline basis functions. While in the second method, the distribution function is assumed to be smooth over the explanatory variable 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 B-spline coefficients 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 have also been modified for quantile grid data where only the percentile range of each response observations are known. Simulations studies have been provided for both complete and quantile grid data. The proposed method has been used to estimate the quantiles of US household income data and North Atlantic hurricane intensity data.