Local linear–additive estimation for multiple nonparametric regressions

Abstract

How to sufficiently use the structure information behind the data is still a challenging issue. In this paper, a local linear–additive estimation and its relevant version are proposed to automatically capture the additive information for general multiple nonparametric regressions. Our method connects two types of local estimators, the local linear (or the local constant) estimator and the local additive estimator. Thus the new estimators can achieve an adaptive fitting between the full model and the local (additive) model, and can adapt to the double additivity: local additivity and global additivity. On the other hand, like the local linear estimator, the new estimators can obtain the optimal convergence rate when the model has no additive structure. Moreover, the new estimators have closed representations and thus make the computation easy and accurate. The theoretical results and simulation studies show that the new approach has a low computational complexity and can significantly improve the estimation accuracy. Also a new theoretical framework is introduced as a foundation of locally and globally connected statistical inference. Based on this framework, the newly defined estimator can be regarded as a projection of the response variable onto full function space with respect to the locally and globally connected norms.