ADAPTIVE ESTIMATION OF THE L2-NORM OF A PROBABILITY DENSITY AND RELATED TOPICS II. UPPER BOUNDS VIA THE ORACLE APPROACH

• 成果类型: Article
• 署名作者: Leanthous, G.; Eorgiadis, A. G.; Epski, O. V.
• 署名单位: Maynooth University; Trinity College Dublin; Centre National de la Recherche Scientifique (CNRS); Aix-Marseille Universite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/25-AOS2503
• 发表日期: 2025
• 页码: 1275-1297
• 关键词: INEQUALITIES

摘要

This is the second part of the research project initiated in Cleanthous, Georgiadis and Lepski (2024a). We deal with the problem of the adaptive estimation of the L-2-norm of a probability density on & Ropf;(d), d >= 1, from independent observations. The unknown density is assumed to be uniformly bounded by unknown constant and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. In Cleanthous, Georgiadis and Lepski (2024a), we have proved that the optimally adaptive estimators do no exist in the considered problem and provided with several lower bounds for the adaptive risk. In this part, we show that these bounds are tight and present the adaptive estimator which is obtained by a data-driven selection from a family of kernel-based estimators. The proposed estimation procedure as well as the computation of its risk are heavily based on new concentration inequalities for decoupled U-statistics of order two established in Section 4. It is also worth noting that all our results are derived from the unique oracle inequality which be of interest.