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Zhou, Harrison H.; Hwang, J. T. Gene

Minimax estimation with thresholding and its application to wavelet analysis. (English) Zbl 1064.62013

Ann. Stat. 33, No. 1, 101-125 (2005).
Summary: 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?
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.

Cited in 2 Documents

MSC:

62C20 Minimax procedures in statistical decision theory
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62J07 Ridge regression; shrinkage estimators (Lasso)
62C12 Empirical decision procedures; empirical Bayes procedures
65C60 Computational problems in statistics (MSC2010)
62G05 Nonparametric estimation

Keywords:

James-Stein estimator; model selection; VisuShrink; SureShrink; BlockJS
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References:

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