Adaptive estimation of the L2-norm of a probability density and related topics I. Lower bounds
G. Cleanthous, A. G. Georgiadis, O. V. Lepski
The Annals of Statistics · 2025 · Volume 53 · Issue 3
Abstract
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.
Keywords
62G05, 62G20, adaptive rate of convergence, anisotropic Nikolskii’s class, estimation of functionals, minimax adaptive estimation, minimax risk
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