
Statistics Papers
Document Type
Journal Article
Date of this Version
2005
Publication Source
The Annals of Statistics
Volume
33
Issue
1
Start Page
184
Last Page
213
DOI
10.1214/009053604000000832
Abstract
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).
Keywords
adaptability, adaptive estimation, shrinking neighborhood, spatially adaptive, superefficiency, wavelets
Recommended Citation
Cai, T., & Low, M. G. (2005). Nonparametric Estimation Over Shrinking Neighborhoods: Superefficiency and Adaptation. The Annals of Statistics, 33 (1), 184-213. http://dx.doi.org/10.1214/009053604000000832
Date Posted: 27 November 2017
This document has been peer reviewed.