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

Understand

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.