Adaptive Non-Parametric Estimation of Smooth Multivariate Functions

Adaptive pointwise estimation of smooth functions f(x) in R d is studied in the white Gaussian noise model of a small intensity ". It is assumed that the Fourier transform of f belongs to a large class of rapidly vanishing functions, but is otherwise unknown. Optimal adaptation in higher dimensions presents several challenges. First, the number of essentially different estimates having a given variance " 2 S increases polynomially, as S d\Gamma1 : Secondly, the set of possible estimators, totally ordered when d = 1; becomes only partially ordered for d ? 1: We demonstrate how these challenges can be met. The first one is to be matched by a meticulous choice of the estimators net. The key to the second problem lies in a new method of spectral majorants introduced in this paper. Extending earlier approach of [12], we restrict ourselves to a family of estimators, rate-efficient in an off-beat case of partially parametric functional classes. A proposed adaptive procedure is shown to ..