A Minimax Property of The Sample Mean in Finite Populations

  • P. J. Bickel
  • E. L. Lehmann
Open Access
Chapter
First Online:
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Consider the problem of estimating the mean of a finite population on the basis of a simple random sample. It was proved by Aggarwal (1954) that the sample mean minimizes the maximum expected squared error divided by the. population variance τ2. Aggarwal also stated, but did not successfully prove, that the sample mean minimizes the maximum expected squared error over the populations satisfying τ2 ≤ M for any fixed positive M. It is the purpose of this paper to give a proof of this second result, and to indicate some generalizations.

Key words and phrases

Finite population simple random sampling minimax estimator means labels stratified sampling sample design 
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References

  1. Aggarwal, Om P. (1959). Bayes and minimax procedures in sampling from finite and infinite populations. Ann. Math. Statist. 30 206–218.MathSciNetMATHCrossRefGoogle Scholar
  2. Blackwell, David and Girshick, M. A. (1954). Theory of Games and Statistical Decisions. Wiley, New York.MATHGoogle Scholar
  3. Girshick, M. A. and Savage, L. J. (1951). Bayes and minimax estimates for quadratic loss functions. Proc. Second Berkeley Symp Math. Statist. Prob. 53–75.Google Scholar
  4. Hodges, J. L., Jr. and Lehmann, E. L. (1981). Minimax estimation in simple random sampling (To appear in Essays in Statistics and Probability in honor of C. R. Rao (P. R. Krishnaiah, ed.). North-Holland.Google Scholar
  5. Kiefer, J. (1957). Invariant minimax sequential estimation and continuous time processes. Ann. Math Statist. 28 573–601.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • P. J. Bickel
    • 1
  • E. L. Lehmann
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

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