Annals of Statistics

Estimation of Normal Means: Frequentist Estimation of Loss

K. L. Lu and James O. Berger

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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

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

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

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