Bayesian Quantile Regression for Single-Index Models

TL;DR

This paper introduces a fully Bayesian approach for single-index models in conditional quantile regression using Gaussian process priors and asymmetric Laplace distribution, with an MCMC algorithm for inference.

Contribution

It develops a novel Bayesian method for single-index quantile regression, integrating Gaussian processes and Bayesian lasso ideas, with a specialized MCMC sampling scheme.

Findings

Bayesian method outperforms frequentist approaches in simulations.

The approach effectively models complex relationships in hurricane data.

The MCMC algorithm efficiently handles the nonparametric link function.

Abstract

Using an asymmetric Laplace distribution, which provides a mechanism for Bayesian inference of quantile regression models, we develop a fully Bayesian approach to fitting single-index models in conditional quantile regression. In this work, we use a Gaussian process prior for the unknown nonparametric link function and a Laplace distribution on the index vector, with the latter motivated by the recent popularity of the Bayesian lasso idea. We design a Markov chain Monte Carlo algorithm for posterior inference. Careful consideration of the singularity of the kernel matrix, and tractability of some of the full conditional distributions leads to a partially collapsed approach where the nonparametric link function is integrated out in some of the sampling steps. Our simulations demonstrate the superior performance of the Bayesian method versus the frequentist approach. The method is further illustrated by an application to the hurricane data.