Statistics Papers

Document Type

Journal Article

Date of this Version

1999

Publication Source

Annals of Statistics

Volume

27

Issue

3

Start Page

898

Last Page

924

DOI

10.1214/aos/1018031262

Abstract

We study wavelet function estimation via the approach of block thresholding and ideal adaptation with oracle. Oracle inequalities are derived and serve as guides for the selection of smoothing parameters. Based on an oracle inequality and motivated by the data compression and localization properties of wavelets, an adaptive wavelet estimator for nonparametric regression is proposed and the optimality of the procedure is investigated. We show that the estimator achieves simultaneously three objectives: adaptivity, spatial adaptivity and computational efficiency. Specifically, it is proved that the estimator attains the exact optimal rates of convergence over a range of Besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point. The estimator is easy to implement, at the computational cost of O(n). Simulation shows that the estimator has excellent numerical performance relative to more traditional wavelet estimators.

Keywords

adaptivity, Besov space, block thresholding, James-Stein estimator, nonparametric regression, oracle inequality, spatial adaptivity, wavelets

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Date Posted: 27 November 2017

This document has been peer reviewed.