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

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.

52 pages

References in corpus (4)

• Estimation of Rényi Entropy and Mutual Information Based on Generalized Nearest-Neighbor Graphs
• Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional
• Nonquadratic estimators of a quadratic functional
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Cited by in corpus (1)

• Minimax Estimation of Quadratic Fourier Functionals