  The paper focuses on the automatic selection of the grouped explanatory
variables in an high-dimensional model, when the model errors are 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
$L_q$-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.
