$\mathbb{L}_{p}$ adaptive estimation of an anisotropic density under independence hypothesis
Résumé
In 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].