Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series B, Pt. 2, pp. 127--148

INCORPORATING INFORMATION ON NEIGHBOURING COEFFICIENTS INTO WAVELET ESTIMATION

By

T. TONY CAI, University of Pennsylvania, Philadelphia, U.S.A

and

BERNARD W. SILVERMAN, University of Bristol, Bristol, U.K

SUMMARY. 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 neighbouring 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.

AMS (1991) subject classification. Primary 62G08; secondary 42C40, 62C20.

Key words and phrases. Adaptivity, Besov space, block thresholding, James-Stein estimator, local adaptivity, nonparametric regression, wavelets, white noise model.

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