Adaptive Bi-Level Variable Selection for Quantile Regression Models with a Diverging Number of Covariates

The paper develops an innovatively adaptive bi-level variable selection methodology for quantile regression models with a diverging number of covariates. Traditional variable selection techniques in quantile regression, such as the lasso and group lasso techniques, offer solutions predominantly for either individual variable selection or group-level selection, but not for both simultaneously. To address this limitation, we introduce an adaptive group bridge approach for quantile regression, to simultaneously select variables at both the group and within-group variable levels. The proposed method offers several notable advantages. Firstly, it adeptly handles the heterogeneous and/or skewed data inherent to quantile regression. Secondly, it is capable of handling quantile regression models where the number of parameters grows with the sample size. Thirdly, via employing an ingeniously designed penalty function, our method surpasses traditional group bridge estimation techniques in identifying important within-group variables with high precision. Fourthly, it exhibits the oracle group selection property, implying that the relevant variables at both the group and within-group levels can be identified with a probability converging to one. Several numerical studies corroborated our theoretical results.