Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures

Abstract

In this paper we develop rate–optimal estimation procedures in the problem of estimating the ${\mathbb{L}}_{\mathit{p}}$–norm, $\mathit{p}\in (1,\infty )$ of a probability density from independent observations. The density is assumed to be defined on ${\mathbb{R}}^{\mathit{d}}$, $\mathit{d}\ge 1$ and to belong to a ball in the anisotropic Nikolskii space. We adopt the minimax approach and construct rate–optimal estimators in the case of integer $\mathit{p}\ge 2$. We demonstrate that, depending on the parameters of the Nikolskii class and the norm index p, the minimax rates of convergence may vary from inconsistency to the parametric $\sqrt{\mathit{n}}$–estimation. The results in this paper complement the minimax lower bounds derived in the companion paper (Goldenshluger and Lepski (2020)).