Adaptive estimation of $\mathbb{L}_2$-norm of a probability density and related topics I. Lower bounds

TL;DR

This paper investigates the limits of adaptively estimating the $\\mathbb{L}_2$-norm of a probability density in multiple dimensions, establishing lower bounds and showing that optimally adaptive estimators do not exist within certain classes.

Contribution

It introduces new lower bounds for adaptive estimation of density functionals and demonstrates the non-existence of optimally adaptive estimators in this context.

Findings

No optimally adaptive estimators exist for the $\mathbb{L}_2$-norm in the considered classes.

Established generic lower bounds for adaptive estimation of density functionals.

Lower bounds are shown to be tight in a companion paper with explicit estimator constructions.

Abstract

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.