Sparse group variable selection based on quantile hierarchical Lasso

The group Lasso is a penalized regression method, used in regression problems where the covariates are partitioned into groups to promote sparsity at the group level [27]. Quantile group Lasso, a natural extension of quantile Lasso [25], is a good alternative when the data has group information and has many outliers and/or heavy tails. How to discover important features that are correlated with interest of outcomes and immune to outliers has been paid much attention. In many applications, however, we may also want to keep the flexibility of selecting variables within a group. In this paper, we develop a sparse group variable selection based on quantile methods which select important covariates at both the group level and within the group level, which penalizes the empirical check loss function by the sum of square root group-wise <italic>L</italic>₁-norm penalties. The oracle properties are established where the number of parameters diverges. We also apply our new method to varying coefficient model with categorial effect modifiers. Simulations and real data example show that the newly proposed method has robust and superior performance.