Gibbs sampling methods for Bayesian quantile regression

Abstract

This paper considers quantile regression models using an asymmetric Laplace distribution from a Bayesian point of view. We develop a simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution. We also discuss some possible extensions of our approach, including the incorporation of a scale parameter, the use of double exponential prior, and a Bayesian analysis of Tobit quantile regression. The proposed methods are illustrated by both simulated and real data.

Quantile Regression Models and Their Applications: A Review

“…However, the ALD is not smooth and thus difficult to maximize its likelihood function. Fortunately, as shown in these studies [14,15], the ALD has various mixture representations. A hierarchical mixture of exponential and normal distributions is utilized to develop algorithms for the QR models [14,15].…”

“…Fortunately, as shown in these studies [14,15], the ALD has various mixture representations. A hierarchical mixture of exponential and normal distributions is utilized to develop algorithms for the QR models [14,15]. These important features of ALD have been generally adopted for likelihood based quantile inference, as well as the Bayesian inference.…”

“…By utilizing this property, under independent data setting, a large number of QRbased statistical models and various associated analysis methods have been investigated in the literature. For example, a likelihood-based goodness-of-fit test has been proposed for QR [6]; Bayesian QR [9] , and the Bayesian estimation procedure for the Tobit QR model with censored data [16,15], have also been developed. Importantly, these two classes of QR inferential methods are not mutually exclusive.…”

Bayesian Analysis of Time-Varying Quantiles Using a Smoothing Spline

“…For non-Gaussian measurement models, it is usually necessary to approximate the non-Gaussian likelihood by the Gaussian likelihood in the previous literature for the MCMC implementation (e.g., Shephard and Pitt (1997), Watanabe and Omori (2004), Kim, Shephard, andChib (1998), Omori, Chib, Shephard, andNakajima (2007)) and in our proposed model, the error distribution is asymmetric double exponential. Noting that it is a normal variance-mean mixture with a generalized inverted Gaussian distribution (e.g., Kotz, Kozubowski, and Podgórski (2001), Tsionas (2003), Kozumi and Kobayashi (2011) and Yue and Rue (2011)), we rewrite…”

A sandwich likelihood correction for Bayesian quantile regression based on the misspecified asymmetric Laplace density

“…The approach is computationally attractive especially due to the location scale mixture normal representation of ALD (see Kozumi and Kobayashi (2011)), and is known to give posterior consistent estimates even if ALD is a misspecification (see Sriram et al 2013). Therefore, the approach has been useful in many applications (e.g.…”