Bayesian Robust Quantile Regression

TL;DR

This paper introduces a Bayesian quantile regression method using the Skew Exponential Power distribution to better handle data with heavy tails, extending traditional models based on the ALD.

Contribution

It proposes a novel SEP distribution-based Bayesian quantile regression framework with adaptive MCMC algorithms, enhancing robustness for fat-tailed data.

Findings

Effective modeling of heavy-tailed data demonstrated

Flexible application to linear and GAM models shown

Improved inference with adaptive MCMC algorithms

Abstract

Traditional Bayesian quantile regression relies on the Asymmetric Laplace distribution (ALD) mainly because of its satisfactory empirical and theoretical performances. However, the ALD displays medium tails and it is not suitable for data characterized by strong deviations from the Gaussian hypothesis. In this paper, we propose an extension of the ALD Bayesian quantile regression framework to account for fat-tails using the Skew Exponential Power (SEP) distribution. Beside having the $\tau$-level quantile as parameter, the SEP distribution has an additional key parameter governing the decay of the tails, making it attractive for robust modeling of conditional quantiles at different confidence levels. Linear and Generalized Additive Models (GAM) with penalized spline are considered to show the flexibility of the SEP in the Bayesian quantile regression context. Lasso priors are considered in both cases to account for shrinking parameters problem when the parameters space becomes wide. To implement the Bayesian inference we propose a new adaptive Metropolis--Hastings algorithm in the linear model and an adaptive Metropolis within Gibbs one in the GAM framework. Empirical evidence of the statistical properties of the proposed SEP Bayesian quantile regression method is provided through several example based on simulated and real dataset.