ScholarlyCommons

Title

Nonparametric Estimation Over Shrinking Neighborhoods: Superefficiency and Adaptation

Author(s)

T. Tony Cai, University of Pennsylvania
Mark G. Low, University of Pennsylvania

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