Adaptive Wavelet Estimation: A Block Thresholding And Oracle Inequality Approach
Tony Cai
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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.
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