  In this paper, we study the problem of pointwise estimation of a multivariate
density. We provide a data-driven selection rule from the family of kernel
estimators and derive for it a pointwise oracle inequality. Using the latter
bound, we show that the proposed estimator is minimax and minimax adaptive over
the scale of anisotropic Nikolskii classes. It is important to emphasize that
our estimation method adjusts automatically to eventual independence structure
of the underlying density. This, in its turn, allows to reduce significantly
the influence of the dimension on the accuracy of estimation (curse of
dimensionality). The main technical tools used in our considerations are
pointwise uniform bounds of empirical processes developed recently in Lepski
[Math. Methods Statist. 22 (2013) 83-99].
