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In the framework of nonparametric multivariate function estimation we areinterested in structural adaptation. We assume that the function to beestimated possesses the single-index structure where neither the link functionnor the index vector is known. We propose a novel procedure that adaptssimultaneously to the unknown index and smoothness of link function. For theproposed procedure, we present a "local" oracle inequality (described by thepointwise seminorm), which is then used to obtain the upper bound on themaximal risk under regularity assumption on the link function. The lower boundon the minimax risk shows that the constructed estimator is optimally rateadaptive over the considered range of classes. For the same procedure we alsoestablish a "global" oracle inequality (under the $ L_r $ norm, $r< \infty $)and study its performance over the Nikol'skii classes. This study shows thatthe proposed method can be applied to estimating functions of inhomogeneoussmoothness.

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