Open Access
February 2005 Minimax estimation with thresholding and its application to wavelet analysis
Harrison H. Zhou, J. T. Gene Hwang
Ann. Statist. 33(1): 101-125 (February 2005). DOI: 10.1214/009053604000000977


Many statistical practices involve choosing between a full model and reduced models where some coefficients are reduced to zero. Data were used to select a model with estimated coefficients. Is it possible to do so and still come up with an estimator always better than the traditional estimator based on the full model? The James–Stein estimator is such an estimator, having a property called minimaxity. However, the estimator considers only one reduced model, namely the origin. Hence it reduces no coefficient estimator to zero or every coefficient estimator to zero. In many applications including wavelet analysis, what should be more desirable is to reduce to zero only the estimators smaller than a threshold, called thresholding in this paper. Is it possible to construct this kind of estimators which are minimax?

In this paper, we construct such minimax estimators which perform thresholding. We apply our recommended estimator to the wavelet analysis and show that it performs the best among the well-known estimators aiming simultaneously at estimation and model selection. Some of our estimators are also shown to be asymptotically optimal.


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Harrison H. Zhou. J. T. Gene Hwang. "Minimax estimation with thresholding and its application to wavelet analysis." Ann. Statist. 33 (1) 101 - 125, February 2005.


Published: February 2005
First available in Project Euclid: 8 April 2005

zbMATH: 1064.62013
MathSciNet: MR2157797
Digital Object Identifier: 10.1214/009053604000000977

Primary: 62G05 , 62J07
Secondary: 62C10 , 62H25

Keywords: BlockJS , James–Stein estimator , Model selection , SureShrink , VisuShrink

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 1 • February 2005
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