Mathematics – Statistics Theory
Scientific paper
2005-04-25
Annals of Statistics 2005, Vol. 33, No. 1, 184-213
Mathematics
Statistics Theory
Published at http://dx.doi.org/10.1214/009053604000000832 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst
Scientific paper
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).
Cai Tony T.
Low Mark G.
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