Wavelet estimation and variable selection for additive partial linear models

Additive partial linear models with nonparametric additive components of heterogeneous smoothness are studied. To achieve optimal rates in large sample situations we use block wavelet penalisation techniques combined with adaptive (group) LASSO procedures for selecting the variables in the linear part and the the additive components in the nonparametric part of the models. Numerical implementations of our procedures for proximal like algorithms are discussed. Large sample properties of the estimates and of the model selection are presented and the results are illustrated with simulated examples and a real data analysis.