  In the framework of nonparametric multivariate function estimation we are
interested in structural adaptation. We assume that the function to be
estimated has the "single-index" structure where neither the link function nor
the index vector is known. We suggest a novel procedure that adapts
simultaneously to the unknown index and smoothness of the link function. For
the proposed procedure, we prove a "local" oracle inequality (described by the
pointwise seminorm), which is then used to obtain the upper bound on the
maximal risk of the adaptive estimator under assumption that the link function
belongs to a scale of H\"{o}lder classes. The lower bound on the minimax risk
shows that in the case of estimating at a given point 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 examine its performance over the Nikol'skii classes.
This study shows that the proposed method can be applied to estimating
functions of inhomogeneous smoothness, that is whose smoothness may vary from
point to point.
