A Study of Blockwise Wavelet Estimates Via Lower Bounds for a Spike Function

Author: SAM EFROMOVICH

Source: Scandinavian Journal of Statistics, Volume 32, Number 1, March 2005 , pp. 133-158(26)

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Abstract:

A blockwise shrinkage is a popular adaptive procedure for non-parametric series estimates. It possesses an impressive range of asymptotic properties, and there is a vast pool of blocks and shrinkage procedures used. Traditionally these estimates are studied via upper bounds on their risks. This article suggests the study of these adaptive estimates via non-asymptotic lower bounds established for a spike underlying function that plays a pivotal role in the wavelet and minimax statistics. While upper-bound inequalities help the statistician to find sufficient conditions for a desirable estimation, the non-asymptotic lower bounds yield necessary conditions and shed a new light on the popular method of adaptation. The suggested method complements and knits together two traditional techniques used in the analysis of adaptive estimates: a numerical study and an asymptotic minimax inference.

Keywords: adaptation; Efromovich–Pinsker shrinkage; James–Stein shrinkage; minimax; regression; Stein shrinkage

Document Type: Research article

DOI: http://dx.doi.org/10.1111/j.1467-9469.2005.00419.x

Affiliations: 1: Department of Mathematics and Statistics, The University of New Mexico

Publication date: 2005-03-01