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February 2012 On Improved Loss Estimation for Shrinkage Estimators
Dominique Fourdrinier, Martin T. Wells
Statist. Sci. 27(1): 61-81 (February 2012). DOI: 10.1214/11-STS380


Let X be a random vector with distribution Pθ where θ is an unknown parameter. When estimating θ by some estimator φ(X) under a loss function L(θ, φ), classical decision theory advocates that such a decision rule should be used if it has suitable properties with respect to the frequentist risk R(θ, φ). However, after having observed X = x, instances arise in practice in which φ is to be accompanied by an assessment of its loss, L(θ, φ(x)), which is unobservable since θ is unknown. A common approach to this assessment is to consider estimation of L(θ, φ(x)) by an estimator δ, called a loss estimator. We present an expository development of loss estimation with substantial emphasis on the setting where the distributional context is normal and its extension to the case where the underlying distribution is spherically symmetric. Our overview covers improved loss estimators for least squares but primarily focuses on shrinkage estimators. Bayes estimation is also considered and comparisons are made with unbiased estimation.


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Dominique Fourdrinier. Martin T. Wells. "On Improved Loss Estimation for Shrinkage Estimators." Statist. Sci. 27 (1) 61 - 81, February 2012.


Published: February 2012
First available in Project Euclid: 14 March 2012

zbMATH: 1330.62287
MathSciNet: MR2953496
Digital Object Identifier: 10.1214/11-STS380

Keywords: conditional inference , linear model , loss estimation , quadratic loss , risk function , robustness , shrinkage estimation , spherical symmetry , SURE , unbiased estimator of loss , uniform distribution on a sphere

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.27 • No. 1 • February 2012
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