Adaptive estimation under single-index constraint in a regression model

The problem of adaptive multivariate function estimation under single-index assumption is studied in the framework of the regression model with random design. We consider the case when both the link function and index vector are unknown. We propose a novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothness of link function by selecting from a family of specific kernel estimators. We establish a pointwise oracle inequality which, in its turn, is used to judge the quality of estimating the entire function (global oracle inequality). Both results are applied to the problems of pointwise and global adaptive estimation over a collection of H\"older and Nikol'skii functional classes.