Wavelets-based non-parametric regression : optimal rate in the sup-norm

Abstract

The model we will study in this paper is a non-parametric regression model on the unit interval [0; 1] with equidistant deterministic grid design. The unknown regression function is assumed to belong to some H?older class with the smoothness parameter > 1 2 . Furthermore the regression function is observed in a Gaussian noise (precise denitions are given in SECTION 2). Ibragimov and Hasminskii [7] have studied a similar kind of model. They considered a more general grid, namely a stochastic grid design, and the class of functions they took was the class of periodic functions on the interval [0; 2?]. Stone [12] obtained optimal rates of convergence where the regression function was assumed to belong to some H?older class of functions. The same kind of model, but with the emphasis on getting the exact constants were studied in Korostelev [8] (for in ] 1 2 ; 1] ) and in Donoho [2] (for > 1 2 ). In both articles the optimal rates where obtained with kernel methods. Our goal in this article is to obtain optimal rates by using wavelet estimators. Recently wavelet estimators were studied in the context of density estimation (cf. Picard and Kerkyacharian [11]) and in estimation the diusion coecient of a diusion process (cf. Genon-Catalot, Laredo and Picard [5]). In Donoho and Johnstone [3] a non-linear wavelet estimator was used for estimating functions with jumps.