Adaptive estimation of the L2-norm of a probability density and related topics II. Upper bounds via the oracle approach
研究从独立观测中自适应估计概率密度L2范数的问题，证明了下界是紧的，并给出通过数据驱动选择核估计器族得到的自适应估计器，结果基于新的二阶解耦U统计量集中不等式。
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 L2-norm of a probability density on Rd, 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 may be of independent interest.