Automatic selection by penalized asymmetric Lq-norm in a high-dimensional model with grouped variables

The paper focuses on the automatic selection of the grouped explanatory variables in a high-dimensional model, when the model blue error is asymmetric. After introducing the model and notations, we define the adaptive group LASSO expectile estimator for which we prove the oracle properties: the sparsity and the asymptotic normality. Afterwards, the results are generalized by considering the asymmetric Lq-norm loss function. The theoretical results are obtained in several cases with respect to the number of variable groups. This number can be fixed or dependent on the sample size n, with the possibility that it is of the same order as n. Note that these new estimators allow us to consider weaker assumptions on the data and on the model errors than the usual ones. Simulation study demonstrates the competitive performance of the proposed penalized expectile regression, especially when the samples size is close to the number of explanatory variables and model errors are asymmetrical. An application on air pollution data is considered.