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

TL;DR

This paper develops rate-optimal estimators for the $L_p$-norm of a probability density in high-dimensional spaces, achieving optimal convergence rates depending on the density's smoothness and the norm parameter.

Contribution

It introduces new minimax estimators for $L_p$-norms on anisotropic Nikolskii spaces, extending previous theoretical bounds and covering a range of convergence behaviors.

Findings

Estimator achieves rate-optimal convergence in various regimes.

Risk asymptotics vary from inconsistency to $ ext{sqrt}(n)$-rate depending on parameters.

Results complement previous minimax lower bounds.

Abstract

In this paper we develop rate--optimal estimation procedures in the problem of estimating the $L_p$--norm, $p\in (0, \infty)$ of a probability density from independent observations. The density is assumed to be defined on $R^d$, $d\geq 1$ 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 $p\geq 2$. We demonstrate that, depending on parameters of Nikolskii's class and the norm index $p$, the risk asymptotics ranges from inconsistency to $\sqrt{n}$--estimation. The results in this paper complement the minimax lower bounds derived in the companion paper \cite{gl20}.