Adaptive non-parametric estimation of smooth multivariate functions

Abstract

Adaptive pointwise estimation of smooth functions fx in Rd is studied in the white Gaussian noise model of a given 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 dierent estimates having a given variance S increases polynomially as Sd Second the set of possible estimators totally ordered when d becomes only partially ordered when d We demonstrate how these challenges can be met The rst 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 we restrict ourselves to a family of estimators rateecient in an obeat case of partially parametric functional classes A proposed adaptive procedure is shown to be asymptotically minimax simultane ously for any ample regular nonparametric family of underlying functions f