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Adaptive Wavelet Estimation: A Block Thresholding And Oracle Inequality Approach

We study wavelet function estimation via the approach of block thresholding and ideal adaptation with oracle. Oracle inequalities are derived to offer insights into the balance and tradeoff between block size and threshold level. Based on the oracle inequalities, an adaptive wavelet method for nonparametric regression is proposed and the optimality of the procedure is discussed. We show that the estimator achieves simultaneously three objectives: adaptivity, spatial adaptivity and high visual quality. Specifically, we show that the estimator attains the exact optimal rates of convergence over a range of Besov classes and perturbed Besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point. Simulation results and generalizations of the method are also discussed. Keywords: James-Stein Estimator; Adaptivity; Wavelet; Block Thresholding; Nonparametric function Estimation; Besov Space. AMS 1991 Subject Classification: Primary 62G07, Secondary ..