The Annals of Statistics

Nonparametric estimation over shrinking neighborhoods: Superefficiency and adaptation

T. Tony Cai and Mark G. Low

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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).

Article information

Ann. Statist., Volume 33, Number 1 (2005), 184-213.

First available in Project Euclid: 8 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G99: None of the above, but in this section
Secondary: 62F12: Asymptotic properties of estimators 62C20: Minimax procedures 62M99: None of the above, but in this section

Adaptability adaptive estimation shrinking neighborhood spatially adaptive superefficiency wavelets


Cai, T. Tony; Low, Mark G. Nonparametric estimation over shrinking neighborhoods: Superefficiency and adaptation. Ann. Statist. 33 (2005), no. 1, 184--213. doi:10.1214/009053604000000832.

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