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].

In this paper, we address the problem of estimating a multidimensional density f by using indirect observations from the statistical model Y = X + ε. Here, ε is a measurement error independent of the random vector X of interest, and having a known density with respect to the Lebesgue measure. Our aim is to obtain optimal accuracy of estimation under Lp-losses when the error ε has a characteristic function with a polynomial decay. To achieve this goal, we first construct a kernel estimator of f which is fully data driven. Then, we derive for it an oracle inequality under very mild assumptions on the characteristic function of the error ε. As a consequence, we get minimax adaptive upper bounds over a large scale of anisotropic Nikolskii classes and we prove that our estimator is asymptotically rate optimal when p ∈ [2, +∞]. Furthermore, our estimation procedure adapts automatically to the possible independence structure of f and this allows us to improve significantly the accuracy of estimation.

International audienceIn this paper, we focus on the problem of a multivariate density estimation under an Lp-loss. We provide a data-driven selection rule from a family of kernel estimators and derive for it Lp-risk oracle inequalities depending on the value of p ≥ 1. The proposed estimator permits us to take into account approximation properties of the underlying density and its independence structure simultaneously. Specifically, we obtain adaptive upper bounds over a scale of anisotropic Nikolskii classes when the smooth- ness is also measured with the Lp-norm. It is important to emphasize that the adaptation to unknown independence structure of the estimated density allows us to improve significantly the accuracy of estimation (curse of di- mensionality). The main technical tools used in our derivation are uniform bounds on the Lp-norms of empirical processes developed in Goldenshluger and Lepski [13]

This paper deals with non-parametric density estimation on R 2 from i.i.d observations. It is assumed that after unknown rotation of the coordinate system the coordinates of the observations are independent random variables whose densities belong to a Hölder class with unknown parameters. The minimax and adaptive minimax theories for this structural statistical model are developed.