We deal with the problem of the adaptive estimation of the $\mathbb{L}_2$-norm of a probability density on $\mathbb{R}^d$, $d\geq 1$, 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 et al (2024) we prove that established lower bounds are tight and provide with explicit construction of adaptive estimators of $\mathbb{L}_2$-norm of the density.

翻译：我们研究从独立观测中自适应估计 $\mathbb{R}^d$（$d\geq 1$）上概率密度 $\mathbb{L}_2$ 范数的问题。假设未知密度一致有界，且属于各向同性/各向异性 Nikolskii 空间中的球的并集。我们将证明，在所考虑的函数类集合上不存在最优的自适应估计量。此外，在一个抽象密度模型的框架下，我们提出了与概率密度任意泛函的自适应估计相关的若干通用下界。这些具有独立意义的结果在现有文献中没有类似物。在配套论文 Cleanthous 等人（2024）中，我们证明了所建立的下界是紧的，并给出了密度 $\mathbb{L}_2$ 范数自适应估计量的显式构造。