May 2022 Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures
Alexander Goldenshluger, Oleg V. Lepski
Author Affiliations +
Bernoulli 28(2): 1155-1178 (May 2022). DOI: 10.3150/21-BEJ1381


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, d1 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 p2. 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)).

Funding Statement

The first author was Supported by the ISF grant No. 361/15.
This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR).


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Alexander Goldenshluger. Oleg V. Lepski. "Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures." Bernoulli 28 (2) 1155 - 1178, May 2022.


Received: 1 March 2021; Revised: 1 June 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388933
zbMATH: 07526579
Digital Object Identifier: 10.3150/21-BEJ1381

Keywords: anisotropic Nikol’skii class , Density estimation , Lp-norm , minimax risk , U-statistics

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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