In this paper we develop rate–optimal estimation procedures in the problem of estimating the –norm, of a probability density from independent observations. The density is assumed to be defined on , 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 . 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 –estimation. The results in this paper complement the minimax lower bounds derived in the companion paper (Goldenshluger and Lepski (2020)).
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).
"Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures." Bernoulli 28 (2) 1155 - 1178, May 2022. https://doi.org/10.3150/21-BEJ1381