Minimax estimation of norms of a probability density: I. Lower bounds

Abstract

The paper deals with the problem of nonparametric estimating the ${\mathbb{L}}_{\mathit{p}}$–norm, $\mathit{p}\in (1,\infty )$, of a probability density on ${\mathbb{R}}^{\mathit{d}}$, $\mathit{d}\ge 1$ from independent observations. The unknown density is assumed to belong to a ball in the anisotropic Nikolskii’s space. We adopt the minimax approach, and derive lower bounds on the minimax risk. In particular, we demonstrate that accuracy of estimation procedures essentially depends on whether p is integer or not. Moreover, we develop a general technique for derivation of lower bounds on the minimax risk in the problems of estimating nonlinear functionals. The proposed technique is applicable for a broad class of nonlinear functionals, and it is used for derivation of the lower bounds in the ${\mathbb{L}}_{\mathit{p}}$–norm estimation.