Variable Selection In Quantile Regression With Adaptive Group Lasso

Quantile regression has been widely used both in theory and in practice,since Koenker first proposed it.Compared with mean regression,quantile regression needs no specific assumption on error distribution,and the objective function is a weighted sum of the absolute deviation,the estimation of regression coefficient is not sensitive to the outliers,thus performs more robust than least squares method,and can more comprehensively depict the influence of explanatory variables on response variables in different quantile level.AS a robust alternative to mean regression analysis?quantile regression is widely used to explore the potential relationship between response variables and explanatory variables.This paper considers a linear model with grouped explanatory variables when the variable dimension p is fixed.In order to automatically select the nonzero variable groups and estimate the regression coefficients,consider the quantile regressioln estimator with adaptive group lasso penalized term,study and prove that this estimator has variable selection consistency,and nonzero coefficients of estimator satisfy asymptotic normality,thus prove that oracle properties of adaptive group lasso quantile regression estimation.In the numerical simulation,the random error obeys the kurtosis distribution and heavy-tailed distribution such as the Cauchy distribution,verify the adaptive estimation of group Lasso quantile(agLasso-Q)group compared with the adaptive Lasso estimation(agLasso-LS)in determining the zero coefficient,with better performance of variable selection.Tuning parameter selection of penalized quantile regression plays an important role in variable selection.For the adaptive group lasso quantile regression estimator,selection of the tuning parameter is different from the widely used selection information criterion such as AIC.BIC,this paper consider a penalized cross validation(PCV)for tuning parameter selection in adaptive group lasso quantile regression,in the form of regarding Schwartz information criterion(SIC)which penalizing the model complexity degree as the loss function of ten cross validation.It is proved that PCV can select true model consistently,and discuss the comparison with other selection criterion.According to the simulation in different quantile points,found that when random error term follow form the kurtosis distribution and heavy-tailed distribution such as t(2)and Cauchy c(0,1),especially at the quantiles of ? = 0.05 and ? = 0.95,PCV performed more better on estimating the regression coefficient,mainly reflected in having smaller mean square error(MSE).