Minimax Estimation with Thresholding and Its Application to Wavelet Analysis

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 base...

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Bibliographic Details
Published in:The Annals of statistics 2005-02, Vol.33 (1), p.101-125
Main Authors: Zhou, Harrison H., J. T. Gene Hwang
Format: Article
Language:English
Subjects:
62C10
62G05
62H25
62J07
Analytical estimating
Bayes estimators
BlockJS
Coefficients
Decision theory
Estimating techniques
Estimation methods
Estimators
Exact sciences and technology
Fourier analysis
James–Stein estimator
Mathematical analysis
Mathematical models
Mathematics
Minimax
model selection
Modeling
Probability and statistics
Sciences and techniques of general use
Statistics
Studies
SureShrink
Threshing
VisuShrink
Wavelet analysis
Wavelet Methods
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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? 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.
ISSN:0090-5364
2168-8966
DOI:10.1214/009053604000000977