Adaptive estimation of the L2-norm of a probability density and related topics I. Lower bounds
研究了从独立观测中自适应估计概率密度L2范数的问题，证明在考虑的函数类上不存在最优自适应估计器，并给出了抽象密度模型下任意泛函自适应估计的通用下界。
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 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 L2–norm of the density.