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 and serv e as guides for the selection of smoothing parameters. Based on an oracle i nequality 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, spatia l adaptivity and computational efficiency. Specifically, it is proved that the estimator attains the exact optimal rates of convergence over a range o f 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 excell ent numerical performance relative to more traditional wavelet estimators.