Improvement on LASSO-type estimator in nonparametric regression

This paper is concerned with nonparametric regression with multi-dimensional, possibly high-dimensional, explanatory variables. A new $ \ell _{1} $ ℓ1-penalisation approach is proposed on the basis of the idea of slicing off waste restriction. The new approach can implement the variable selection and estimate the regression function simultaneously. The approach also develops the consistency of the variable selection and the asymptotic theory of regression estimator, thus highlighting the advantages of the proposed penalisation approach. Simulation results show that the proposed approaches for variable selection work efficiently, and the associated regression estimators perform well. Applications to some real data sets also reveal that the proposed methods yield reasonable and stable solutions for nonparametric regression problems.