Lepski's Method and Adaptive Estimation of Nonlinear Integral Functionals of Density

TL;DR

This paper develops an adaptive estimation method for nonlinear integral functionals of a density, extending Lepski's method to handle complex estimators like higher order U-statistics.

Contribution

It introduces a modified Lepski's method suitable for nonlinear functionals and provides a way to control moments of minimax estimators, establishing optimal adaptation rates.

Findings

Extended Lepski's method for nonlinear functionals

Controlled higher order moments of minimax estimators

Proved optimality of adaptive rates

Abstract

We study the adaptive minimax estimation of non-linear integral functionals of a density and extend the results obtained for linear and quadratic functionals to general functionals. The typical rate optimal non-adaptive minimax estimators of "smooth" non-linear functionals are higher order U-statistics. Since Lepski's method requires tight control of tails of such estimators, we bypass such calculations by a modification of Lepski's method which is applicable in such situations. As a necessary ingredient, we also provide a method to control higher order moments of minimax estimator of cubic integral functionals. Following a standard constrained risk inequality method, we also show the optimality of our adaptation rates.