Incorporating Information on Neighboring Coeecients into Wavelet Estimation

In standard wavelet methods, the empirical wavelet coeecients are thresholded term by term, on the basis of their individual magnitudes. Information on other coeecients has no innuence on the treatment of particular coeecients. We propose a wavelet shrinkage method that incorporates information on neighboring coeecients into the decision making. The coeecients are considered in overlapping blocks; the treatment of coeecients in the middle of each block depends on the data in the whole block. The asymptotic and numerical performances of two particular versions of the estimator are investigated. We show that, asymptotically, one version of the estimator achieves the exact optimal rates of convergence over a range of Besov classes for global estimation, and attains adaptive minimax rate for estimating functions at a point. In numerical comparisons with various methods, both versions of the estimator perform excellently.