Open Access
June 1999 Adaptive wavelet estimation: a block thresholding and oracle inequality approach
T. Tony Cai
Ann. Statist. 27(3): 898-924 (June 1999). DOI: 10.1214/aos/1018031262


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.


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T. Tony Cai. "Adaptive wavelet estimation: a block thresholding and oracle inequality approach." Ann. Statist. 27 (3) 898 - 924, June 1999.


Published: June 1999
First available in Project Euclid: 5 April 2002

zbMATH: 0954.62047
MathSciNet: MR1724035
Digital Object Identifier: 10.1214/aos/1018031262

Primary: 62G07
Secondary: 62G20

Keywords: Adaptivity , Besov space , block thresholding , James-Stein estimator , Nonparametric regression , Oracle inequality , spatial adaptivity , Wavelets

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • June 1999
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