The Annals of Statistics

Estimation of Normal Means: Frequentist Estimation of Loss

K. L. Lu and James O. Berger

Full-text: Open access

Abstract

In estimation of a $p$-variate normal mean with identity covariance matrix, Stein-type estimators can offer significant gains over the $\operatorname{mle}$ in terms of risk with respect to sum of squares error loss. Their maximum risk is still equal to $p$, however, which will typically be their "reported loss." In this paper we consider use of data-dependent "loss estimators." Two conditions that are attractive for such a loss estimator are that it be an improved loss estimator under some scoring rule and that it have a type of frequentist validity. Loss estimators with these properties are found for several of the most important Stein-type estimators. One such estimator is a generalized Bayes estimator, and the corresponding loss estimator is its posterior expected loss. Thus Bayesians and frequentists can potentially agree on the analysis of this problem.

Article information

Source
Ann. Statist. Volume 17, Number 2 (1989), 890-906.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176347149

Digital Object Identifier
doi:10.1214/aos/1176347149

Mathematical Reviews number (MathSciNet)
MR994274

Zentralblatt MATH identifier
0714.62003

JSTOR
links.jstor.org

Subjects
Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62C15: Admissibility

Keywords
Estimated loss communication loss communication risk Stein estimation generalized Bayes estimator posterior variance

Citation

Lu, K. L.; Berger, James O. Estimation of Normal Means: Frequentist Estimation of Loss. Ann. Statist. 17 (1989), no. 2, 890--906. doi:10.1214/aos/1176347149. http://projecteuclid.org/euclid.aos/1176347149.


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