Estimation adaptative dans le mod\`ele single-index par l'approche d'oracle

In the framework of nonparametric multivariate function estimation we are interested in structural adaptation. We assume that the function to be estimated possesses the single-index structure where neither the link function nor the index vector is known. We propose a novel procedure that adapts simultaneously to the unknown index and smoothness of link function. For the proposed procedure, we present a "local" oracle inequality (described by the pointwise seminorm), which is then used to obtain the upper bound on the maximal risk under regularity assumption on the link function. The lower bound on the minimax risk shows that the constructed estimator is optimally rate adaptive over the considered range of classes. For the same procedure we also establish a "global" oracle inequality (under the $L_r$ norm, $r< \infty$) and study its performance over the Nikol'skii classes. This study shows that the proposed method can be applied to estimating functions of inhomogeneous smoothness.