## The Annals of Statistics

### Adaptive wavelet estimation: a block thresholding and oracle inequality approach

T. Tony Cai

#### 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.

#### Article information

Source
Ann. Statist., Volume 27, Number 3 (1999), 898-924.

Dates
First available in Project Euclid: 5 April 2002

https://projecteuclid.org/euclid.aos/1018031262

Digital Object Identifier
doi:10.1214/aos/1018031262

Mathematical Reviews number (MathSciNet)
MR1724035

Zentralblatt MATH identifier
0954.62047

Subjects
Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

#### Citation

Cai, T. Tony. Adaptive wavelet estimation: a block thresholding and oracle inequality approach. Ann. Statist. 27 (1999), no. 3, 898--924. doi:10.1214/aos/1018031262. https://projecteuclid.org/euclid.aos/1018031262

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