Adaptive estimation in the single-index model

The problem of adaptive estimation of a multivariate function under the single-index constrains when both the link function and index vector are unknown is investigated. 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 as well (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.