Incorporating Information on Neighboring Coefficients into Wavelet Estimation

In standard wavelet methods, the empirical wavelet coefficients are thresholded term by term, on the basis of their individual magnitudes. Information on other coefficients has no influence on the treatment of particular coefficients. We propose and investigate a wavelet shrinkage method that incorporates information on neighboring coefficients into the decision making. The coefficients are considered in overlapping blocks; the treatment of coefficients in the middle of each block depends on the data in the whole block. Both the asymptotic and numerical performances of two particular versions of the estimator are considered. In numerical comparisons with various methods, both versions of the estimator perform excellently; on the theoretical side, we show that one of the versions achieves the exact optimal rates of convergence over a range of Besov classes.