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Non-Optimality of Preliminary-Test Estimators for the Mean of a Multivariate Normal Distribution
Open accessAbstract
Estimation-preceded-by-testing is studied in the context of estimating the mean vector of a multivariate normal distribution with quadratic loss. It is shown that although there are parameter values for which the risk of a preliminary-test estimator is less than that of the usual estimator, there are also values for which its risk exceeds that of the usual estimator, and that it is dominated by the positive-part version of the Stein-James estimator. The results apply to preliminary-test estimators corresponding to any linear hypothesis concerning the mean vector, e.g., an hypothesis in a regression model. The case in which the covariance matrix of the multi-normal distribution is known up to a multiplicative constant and the case in which it is completely unknown are treated.
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Almost 30 years of empirical Bayes research show that modeling approaches that shrink a coefficient part of the way back to zero can dominate all-or-nothing methods.
“by now we have almost 30 years of good empirical Bayes research (e.g., Sclove, Morris, Radhakrishna, 1972; Copas, 1983) to show that modeling approaches that shrink a given coefficient part of the way back to zero can dominate all-or-nothing methods.”
From Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors (1999) · cites this paper as Sclove, Morris, Radhakrishna, 1972
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Research works printed but not linked: 3
- [1][1] ANDERsON, T. W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley, New York. -
- [2][2] JAMEs, W. and STEIN, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1 361-379. Univ. of California Press. -
- [3][3] STEIN, C. (1966). An approach to the recovery of interblock information in balanced incomplete block designs. Research Papers in Statistics: Festschrift for J. Neyman, (F. N. David ed.). 351-366. Wiley, New York. DEPARTMENT OF STATISTICS STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305
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Hoeting, J. A., Madigan, D., Raftery, A. E., & Volinsky, C. T. (1999) · Statistical Science · doi:10.1214/ss/1009212519 · Cited by 4,118
Metadata
- Authors
- Sclove, S. L., Morris, C., & Radhakrishnan, R.
- ORCID
- Sclove, Stanley L.
- Venue
- The Annals of Mathematical Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1972-10
- Volume
- 43
- Issue
- 5
- Pages
- 1481-1490
- ISSN
- 0003-4851 (ISSN-L)
- OpenAlex
- W2026073235
- Cited by
- 180
- References
- 0
- Citations by year
- 2026: 2 · 2025: 1 · 2024: 2 · 2022: 1 · 2021: 1 · 2018: 3 · 2017: 2 · 2016: 3 · 2015: 3 · 2014: 2 · 2013: 3 · 2012: 3
- FWCI
- 6.92
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- The Annals of Mathematical Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- preliminary test estimation, multivariate normal mean, squared error loss, linear hypotheses, risk comparison, unknown covariance matrix, regression models
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