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

Abstract

In this paper, we focus on the problem of a multivariate density estimation under an $\mathbb{L}_{p}$-loss. We provide a data-driven selection rule from a family of kernel estimators and derive for it $\mathbb{L}_{p}$-risk oracle inequalities depending on the value of $p\geq1$. 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 smoothness is also measured with the $\mathbb{L}_{p}$-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 dimensionality). The main technical tools used in our derivation are uniform bounds on the $\mathbb{L}_{p}$-norms of empirical processes developed in Goldenshluger and Lepski [13].