The Annals of Statistics

Minimax estimation with thresholding and its application to wavelet analysis

Harrison H. Zhou and J. T. Gene Hwang

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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.

Article information

Ann. Statist. Volume 33, Number 1 (2005), 101-125.

First available in Project Euclid: 8 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62J07: Ridge regression; shrinkage estimators
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62H25: Factor analysis and principal components; correspondence analysis

James–Stein estimator model selection VisuShrink SureShrink BlockJS


Zhou, Harrison H.; Hwang, J. T. Gene. Minimax estimation with thresholding and its application to wavelet analysis. Ann. Statist. 33 (2005), no. 1, 101--125. doi:10.1214/009053604000000977.

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