In recent years, variable selection based on penalty likelihood methods has aroused great concern. Based on the Gibbs sampling algorithm of asymmetric Laplace distribution, this paper considers the quantile regression with adaptive Lasso and Lasso penalty from a Bayesian point of view. Under the non-Bayesian and Bayesian framework, several regularization quantile regression methods are systematically compared for error terms with different distributions and heteroscedasticity. Under the error

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... rm of asymmetric Laplace distribution, statistical simulation results show that the Bayesian regularized quantile regression is superior to other distributions in all quantiles. And based on the asymmetric Laplace distribution, the Bayesian regularized quantile regression approach performs better than the non-Bayesian approach in parameter estimation and prediction. Through real data analyses, we also confirm the above conclusions. tile regression, which is the natural analog of R2 statistic of least squares regression [2]. In 2001, Yu and Moyeed first proposed a Bayesian quantile regression model that the error follows an asymmetric Laplace ( AL) distribution, and proved the maximization of likelihood-based inference used independently distributed asymmetric Laplace densities was equivalent to the minimization of the loss function [3]. In 2010, Hewson and Yu suggested quantile regression model for binary data within the Bayesian framework [4]. In 2011, Reich et al. introduced Bayesian spatial quantile regression model [5]. In 2013, Sriram et al. showed that the misspecified likelihood in the ALD approach still leads to consistent results [6]. In 2009, Kozumi and Kobayashi built a more efficient Gibbs sampler for fitted the quantile regression model based on a location-scale mixture of the asymmetric Laplace distribution to draw samples from the posterior distribution [7]. In 2012, Khare and Hobert proved that this new sampling algorithm converges at a geometric rate [8]. In 2015, Sriram proposed a correction to