Adaptive Non-Parametric Estimation of Smooth Multivariate Functions

Abstract

Adaptive pointwise estimation of smooth functions f(x) in R is studied in the white Gaussian noise model of a given intensity " ! 0. 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 " S increases polynomially, : Second, the set of possible estimators, totally ordered when d = 1; becomes only partially ordered when 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 solving the second problem lies in a new method of spectral majorants introduced in this paper. Extending our earlier approach used in [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 be asymptotically minimax, simultaneously for any ample regular non-parametric family of underlying functions f