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

Alexander Goldenshluger, Oleg V. Lepski

Research output: Contribution to journalArticlepeer-review


The paper deals with the problem of nonparametric estimating the Lp –norm, p ∈ (1, ∞), of a probability density on Rd, d ≥ 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 Lp –norm estimation.

Original languageEnglish
Pages (from-to)1120-1154
Number of pages35
Issue number2
StatePublished - May 2022

Bibliographical note

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  • Anisotropic Nikolskii’s class
  • Best approximation
  • Estimation of nonlinear functionals
  • Minimax estimation
  • Minimax risk

ASJC Scopus subject areas

  • Statistics and Probability


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