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

Alexander Goldenshluger, Oleg V. Lepski

Research output: Contribution to journalArticlepeer-review


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)).

Original languageEnglish
Pages (from-to)1155-1178
Number of pages24
Issue number2
StatePublished - May 2022

Bibliographical note

Funding Information:
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).

Funding Information:
The first author was Supported by the ISF grant No. 361/15.

Publisher Copyright:
© 2022.


  • Anisotropic Nikol’skii class
  • Density estimation
  • L-norm
  • Minimax risk
  • U-statistics

ASJC Scopus subject areas

  • Statistics and Probability


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