We study the problem of adaptive estimation of a multivariate function under the single-index constrains when both the link function and index vector are unknown. We propose a novel estimation procedure adapting simultaneously to the unknown index vector and the smoothness of the link function by selection from a family of specific kernel estimators. This procedure allows to establish a pointwise oracle inequality which can be used to judge the quality of estimation under the L_r-norm losses. The obtained oracle inequality allows adaptation with respect to a scale of Nikol'skii (Besov) classes B_{p, \infty}^s, where s is the unknown regularity of link function and p is the index of L_p-norm involved in the class definition. This paper therefore covers non-treated in the literature on structural adaptation and, correspondingly by the recent Goldenshluger-Lepski procedure, case of r > p. We also provide the lower bounds for the considered class of functions possessing the single index structure.
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