Bayesian Quantile Regression Using Random B-spline Series Prior

Abstract

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 $\xi_1$ and $\xi_2$ not depending on the prediction variables. In a Bayesian approach, a prior is put on quantile functions by putting prior distributions on $\xi_1$ and $\xi_2$. The monotonicity constraint on the curves $\xi_1$ and $\xi_2$ 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.