Bayesian Quantile Regression Using Random B-spline Series Prior

TL;DR

This paper introduces a Bayesian quantile regression method using random B-spline series priors, enabling flexible modeling of quantile functions with monotonicity constraints, and demonstrates its effectiveness through simulations and real data applications.

Contribution

The paper proposes a novel Bayesian quantile regression approach employing spline basis expansion with shape constraints, improving flexibility and interpretability over existing methods.

Findings

The method performs well in simulation studies compared to Gaussian process priors.

It effectively models complex quantile functions with monotonicity constraints.

Applications to hurricane and population data demonstrate practical utility.

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.