  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.
