Adaptive model selection in a high-dimension multiphase quantile regression

We propose a general adaptive LASSO method for an high-dimension quantile regression, which is very interesting when we know nothing about the first two moments of the model error. We first propose that the obtained estimators satisfy the oracle properties, which involves the relevant variable selection without crossing through the hypothesis test. Next, we study the proposed method when the (multiphase) model changes to unknown observations called change-points. Convergence rates of the change-points and of the regression parameters estimators in each phase are found. The sparsity of the adaptive LASSO quantile estimators of the regression parameters is not affected by the change-points estimation. If the phases number is unknown, a consistent criterion is proposed. Numerical studies by Monte Carlo simulations show the performance of the proposed method, compared to other existing methods in the literature, in models with one or multi-phases.