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December 2004 Hybrid shrinkage estimators using penalty bases for the ordinal one-way layout
Rudolf Beran
Ann. Statist. 32(6): 2532-2558 (December 2004). DOI: 10.1214/009053604000000652


This paper constructs improved estimators of the means in the Gaussian saturated one-way layout with an ordinal factor. The least squares estimator for the mean vector in this saturated model is usually inadmissible. The hybrid shrinkage estimators of this paper exploit the possibility of slow variation in the dependence of the means on the ordered factor levels but do not assume it and respond well to faster variation if present. To motivate the development, candidate penalized least squares (PLS) estimators for the mean vector of a one-way layout are represented as shrinkage estimators relative to the penalty basis for the regression space. This canonical representation suggests further classes of candidate estimators for the unknown means: monotone shrinkage (MS) estimators or soft-thresholding (ST) estimators or, most generally, hybrid shrinkage (HS) estimators that combine the preceding two strategies. Adaptation selects the estimator within a candidate class that minimizes estimated risk. Under the Gaussian saturated one-way layout model, such adaptive estimators minimize risk asymptotically over the class of candidate estimators as the number of factor levels tends to infinity. Thereby, adaptive HS estimators asymptotically dominate adaptive MS and adaptive ST estimators as well as the least squares estimator. Local annihilators of polynomials, among them difference operators, generate penalty bases suitable for a range of numerical examples. In case studies, adaptive HS estimators recover high frequency details in the mean vector more reliably than PLS or MS estimators and low frequency details more reliably than ST estimators.


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Rudolf Beran. "Hybrid shrinkage estimators using penalty bases for the ordinal one-way layout." Ann. Statist. 32 (6) 2532 - 2558, December 2004.


Published: December 2004
First available in Project Euclid: 7 February 2005

zbMATH: 1068.62078
MathSciNet: MR2153994
Digital Object Identifier: 10.1214/009053604000000652

Primary: 62J07

Keywords: Adaptation , estimated risk , local annihilator , monotone shrinkage , One-way layout , ordinal factor , penalized least squares , soft-thresholding

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 6 • December 2004
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