ADAPTIVE ESTIMATION OF THE L2-NORM OF A PROBABILITY DENSITY AND RELATED TOPICS I. LOWER BOUNDS

• 成果类型: Article
• 署名作者: Leanthous, C.; 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-AOS2502
• 发表日期: 2025
• 页码: 1257-1274
• 关键词: integral functionals; pointwise; sharp; NORM

摘要

We deal with the problem of the adaptive estimation of the L-2-norm of a probability density on R-d, d >= 1, from independent observations. The unknown density is assumed to be uniformly bounded and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. We will show that the optimally adaptive estimators over the collection of considered functional classes do no exist. Also, in the framework of an abstract density model we present several generic lower bounds related to the adaptive estimation of an arbitrary functional of a probability density. These results having independent interest have no analogue in the existing literature. In the companion paper (Cleanthous, Georgiadis and Lepski (2024)), we prove that established lower bounds are tight and provide with explicit construction of adaptive estimators of L-2-norm of the density.