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 Lp–norm, p∈(1,∞) of a probability density from independent observations. The density is assumed to be defined on Rd, d≥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 p≥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 n–estimation. The results in this paper complement the minimax lower bounds derived in the companion paper (Goldenshluger and Lepski (2020)).