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Adaptive wavelet estimation: A block thresholding and oracle inequality approach. (English) Zbl 0954.62047

Summary: 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.

MSC:

62G08 Nonparametric regression and quantile regression
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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