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.
Bibliographical noteFunding 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).
The first author was supported by the ISF grant No. 361/15.
- Anisotropic Nikolskii’s class
- Best approximation
- Estimation of nonlinear functionals
- Minimax estimation
- Minimax risk
ASJC Scopus subject areas
- Statistics and Probability