Open Access
February 2005 Nonparametric estimation over shrinking neighborhoods: Superefficiency and adaptation
T. Tony Cai, Mark G. Low
Ann. Statist. 33(1): 184-213 (February 2005). DOI: 10.1214/009053604000000832


A theory of superefficiency and adaptation is developed under flexible performance measures which give a multiresolution view of risk and bridge the gap between pointwise and global estimation. This theory provides a useful benchmark for the evaluation of spatially adaptive estimators and shows that the possible degree of superefficiency for minimax rate optimal estimators critically depends on the size of the neighborhood over which the risk is measured.

Wavelet procedures are given which adapt rate optimally for given shrinking neighborhoods including the extreme cases of mean squared error at a point and mean integrated squared error over the whole interval. These adaptive procedures are based on a new wavelet block thresholding scheme which combines both the commonly used horizontal blocking of wavelet coefficients (at the same resolution level) and vertical blocking of coefficients (across different resolution levels).


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T. Tony Cai. Mark G. Low. "Nonparametric estimation over shrinking neighborhoods: Superefficiency and adaptation." Ann. Statist. 33 (1) 184 - 213, February 2005.


Published: February 2005
First available in Project Euclid: 8 April 2005

zbMATH: 1064.62032
MathSciNet: MR2157801
Digital Object Identifier: 10.1214/009053604000000832

Primary: 62G99
Secondary: 62C20 , 62F12 , 62M99

Keywords: Adaptability , adaptive estimation , shrinking neighborhood , spatially adaptive , superefficiency , Wavelets

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 1 • February 2005
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