Incorporate Information on Neighboring Coefficients into Wavelet Estimation
Sankhya 63, 127-148, (2001).

Tony Cai and Bernard W. Silverman

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• Abstract : 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.

• Paper : postscript file or pdf file.

• More simulation results on the estimators, NeighBlock and NeighCoeff, can be found in this report:
  • postscript file
  • pdf file.

• SPLUS scripts implementing the estimators can be downloaded here:
  • NeighBlock
  • NeighCoeff
• Help files for the functions NeighBlock and NeighCoeff:
  • Help file for NeighBlock
  • Help file for NeighCoeff
  • Check here if you are not sure where to put the help files.

• Other related papers:

  Cai, T. (1999).
  Adaptive wavelet estimation: a block thresholding and oracle inequality approach.
  The Annals of Statistics 27, 898-924.

  Cai, T. (2002).
  On block thresholding in wavelet regression: Adaptivity, block size, and threshold level.
  Statistica Sinica 12, 1241-1273.

  Cai, T. (2002).
  On adaptive wavelet estimation of a derivative and other related linear inverse problems.
  J. Statistical Planning and Inference 108, 329-349.