Adaptive estimation of the $\mathbb{L}_2$-norm of a probability density and related topics II. Upper bounds via the oracle approach

TL;DR

This paper develops a data-driven, adaptive kernel estimator for the $\,\mathbb{L}_2$-norm of a probability density, achieving tight bounds and utilizing new concentration inequalities for decoupled U-statistics.

Contribution

It introduces an adaptive estimation method for the $\,\mathbb{L}_2$-norm that matches lower bounds, based on novel concentration inequalities and oracle inequalities.

Findings

Proposed estimator is optimally adaptive within the considered setting.

New concentration inequalities for decoupled U-statistics are established.

The estimator's risk bounds are shown to be tight and theoretically justified.

Abstract

This is the second part of the research project initiated in Cleanthous et al (2024). 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 by unknown constant and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. In Cleanthous et al (2024) 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.