Nonparametric Function Estimation Over Shrinking Neighborhoods: Superefficiency And Adaptation
Tony Cai and Mark Low
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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).
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- 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. & Silverman, B.W. (2001).
Incorporating information on neighboring coefficients into wavelet estimation.
Sankhya 63, 127-148.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.