Open Access
June 1998 Block threshold rules for curve estimation using kernel and wavelet methods
Peter Hall, Gérard Kerkyacharian, Dominique Picard
Ann. Statist. 26(3): 922-942 (June 1998). DOI: 10.1214/aos/1024691082

Abstract

Motivated by recently developed threshold rules for wavelet estimators, we suggest threshold methods for general kernel density estimators, including those of classical Rosenblatt–Parzen type. Thresholding makes kernel methods competitive in terms of their adaptivity to a wide variety of aberrations in complex signals. It is argued that term-by-term thresholding does not always produce optimal performance, since individual coefficients cannot be estimated sufficiently accurately for reliable decisions to be made. Therefore, we suggest grouping coefficients into blocks and making simultaneous threshold decisions about all coefficients within a given block. It is argued that block thresholding has a number of advantages, including that it produces adaptive estimators which achieve minimax-optimal convergence rates without the logarithmic penalty that is sometimes associated with term-by-term thresholding. More than this, the convergence rates are achieved over large classes of functions with discontinuities, indeed with a number of discontinuities that diverges polynomially fast with sample size. These results are also established for block thresholded wavelet estimators, which, although they can be interpreted within the kernel framework, are often most conveniently constructed in a slightly different way.

Citation

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Peter Hall. Gérard Kerkyacharian. Dominique Picard. "Block threshold rules for curve estimation using kernel and wavelet methods." Ann. Statist. 26 (3) 922 - 942, June 1998. https://doi.org/10.1214/aos/1024691082

Information

Published: June 1998
First available in Project Euclid: 21 June 2002

zbMATH: 0929.62040
MathSciNet: MR1635418
Digital Object Identifier: 10.1214/aos/1024691082

Subjects:
Primary: 62G07
Secondary: 62G20

Keywords: Adaptivity , bias , convergence rate , Density estimation , minimax , Nonparametric regression , smoothing parameter , variance

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • June 1998
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