Abstract
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 term 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.
Highlights
Since the pioneering work by Koenker and Bassett in 1978, quantile regression (QR) has been deeply studied and widely applied to descript the elaborate relationship between the dependent variable and predictors [1]
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
Bayesian quantile regression with adaptive Lasso penalty (BQR-AL) is based on different penalty parameters are applied to different regression coefficients
Summary
Since the pioneering work by Koenker and Bassett in 1978, quantile regression (QR) has been deeply studied and widely applied to descript the elaborate relationship between the dependent variable and predictors [1]. 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 2008, Park and Casella studied the Lasso penalty from the Bayesian angle, and proposed that the hierarchical model can be effectively solved by the Gibbs sampler, thereby introducing the regularization method [14]. In 2010, Li et al studied the regularization method in quantile regression from the perspective of Bayesian and proposed to set the prior distribution of parameters to Laplace prior, and use Gibbs sampler to sampling Bayesian Lasso quantile regression [15]. In 2018, Adlouni et al showed that a regularized quantile regression model with B-Splines based on five penalties (Lasso, Ridge, SCAD0, SCAD1 and SCAD2) in Bayesian framework [17]. The prostate cancer data sets are used to illustrate the advantages and disadvantages of these two approaches
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