Improved Confidence Statements for the Usual Multivariate Normal Confidence Set

  • Christian Robert
  • George Casella

Abstract

The usual multivariate normal confidence set has reported confidence 1—α, which is equal to its coverage probability. If we take a decision theoretic view, and attempt to estimate the coverage, we find that 1— α is an inadmissible estimator in more than four dimensions. We establish this fact and, moreover, exhibit adaptive confidence estimators that appear to dominate 1— α. These new confidence estimators are developed through empirical Bayes arguments and approximations. They allow us to attach confidence that is uniformly greater than 1— α. We provide necessary conditions, and strong numerical evidence to support our domination claims.

Keywords

Covariance Stein 

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References

  1. Berger, J. 0. (1985x). Statistical Decision Theory and Bayesian Analysis; Springer (2nd edition).Google Scholar
  2. Berger, J. 0. (1985b). The frequentist viewpoint and conditioning. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer ( L. LeCam and R. Olshen, eds.). Wadsworth, Monterey, CA.Google Scholar
  3. Berger, J. 0. (1988). Discussion of “Conditionally acceptable frequentist solutions.” In Statistical Decision Theory and Related Topics, Vol. IV (S. Gupta and J. Berger, eds.). Springer-Verlag, NY.Google Scholar
  4. Berger, J. O. (1990). On the inadmissibility of unbiased estimators. Stat. Prob. Letters 9 (5), 381–384.MATHCrossRefGoogle Scholar
  5. Berger, J. 0. and Robert, C. (1990). Subjective hierarchical Bayes estimation of a normal mean: on the frequentist interface. The Annals of Statistics 18, 617–651.MathSciNetMATHCrossRefGoogle Scholar
  6. Berger, J. O. and Sellke, T. (1987). Testing a point null hypothesis: the irreconcilability of p-values and evidence. JASA 82, 112–122.MathSciNetMATHGoogle Scholar
  7. Berger, J. O. and Wolpert, R. (1989). The Likelihood Principle. IMS Monograph, Series 6, 2nd edition.Google Scholar
  8. Brown, L. D. (1988). The differential inequality of a statistical estimation problem. In Statistical Decision Theory and Related Topics, Yol. IY ( S. Gupta and J. Berger, eds.). Springer-Verlag, NY.Google Scholar
  9. Brown, L. D. (1967). The conditional level of Student’s test. Ann. Statist. 38, 1068–1071.MATHCrossRefGoogle Scholar
  10. Brown, L. D. (1975). Estimation with incompletely specified loss functions, J. Amer. Statist. Assoc. 70, 417–26.MathSciNetMATHCrossRefGoogle Scholar
  11. Brown, L. D. and Hwang, J. T. (1991). Admissibility of confidence estimators. Proceedings of the 1990 Taipei 5ymposiurn in Statistics, June 28–30, 1990. M. T. Chao and P. E. Cheng (eds.). Institute of Statistical Science, Academics Sinica Taiwan.Google Scholar
  12. Casella, G. (1990). Conditional inference from confidence sets. Technical Report BU-1001¬M, Cornell University, Ithaca, NY. To appear in IMS Lecture Notes Series, Yolume in Honor of D. Basu.Google Scholar
  13. George, E. I. and Casella, G. (1990). Empirical Bayes confidence estimation. Technical Report BU-1062-M, Cornell University, Ithaca, NY.Google Scholar
  14. Efron, B. and Morris, C. (1973). Stein estimation and its competitor - an Empirical Bayes approach. JASA 68, 117–130.MathSciNetMATHGoogle Scholar
  15. Hwang, J. T. and Brown, L. D. (1991). The estimated confidence approach under the validity constraint criterion. Ann. Stat. 19, 1964–1977.MathSciNetMATHCrossRefGoogle Scholar
  16. Hwang, J. T. and Casella, G. (1982). Minimax confidence sets for the mean of a multivariate normal distribution. Ann. Stat. 10, 868–881.MathSciNetMATHCrossRefGoogle Scholar
  17. Hwang, J. T., Casella, G., Robert, C., Wells, M. T., Farrell, R. (1992). Estimation of accuracy in testing. Ann. Stat. 20, 490–509.MathSciNetMATHCrossRefGoogle Scholar
  18. Hwang, J. T. and Ullah, A. (1989). Confidence seta centered at James-Stein estimators. Technical Report, Mathematics Department, Cornell University.Google Scholar
  19. Johnstone, I. (1988). On the inadmissibility of Stein’s unbiased estimate of loss. In Statistical Decision Theory and Related Topics, Yol. IY (S. Gupta and J. Berger, ede.). Springer-Verlag, NY.Google Scholar
  20. Kass, R. and Steffey, D. (1989). Parametric Empirical Bayes Models. JASA 86, 717–726.MathSciNetGoogle Scholar
  21. Lu, K. and Berger, J. (1989x). Estimated confidence procedures for multivariate normal means. J. 5tat. Plans. Inf. 23, 1–19.MathSciNetMATHCrossRefGoogle Scholar
  22. Lu, K. and Berger, J. (1989b). Estimation of normal means: Frequentists estimators of loss. Ann. Stat. 17, 890–907.MathSciNetMATHCrossRefGoogle Scholar
  23. Morris, C. (1983). Parametric Empirical Bayes Inference: Theory and Applications. JASA 78, 47–65.MATHGoogle Scholar
  24. Robert, C. and Casella, G. (1990). Improved confidence sets for spherically symmetric distributions. J. Multivariate Analysis 32 (1), 84–94.MathSciNetMATHCrossRefGoogle Scholar
  25. Robinson, G. (1979a). Conditional properties of statistical procedures. Ann. Stat. 7, 742–755.MATHCrossRefGoogle Scholar
  26. Robinson, G. K. (1979b). Conditional properties of statistical procedures for location and scale parameters. Ann. Stat. 7, 756–771.MATHCrossRefGoogle Scholar
  27. Rukhin, A. (1988). Estimated loss and admissible loss estimators. In Statistical Decision Theory and Related Topics, Yol. IV ( S. Gupta and J. Berger, eds.). Springer-Verlag, NY.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Christian Robert
    • 1
  • George Casella
    • 2
  1. 1.Université Paris VIFrance
  2. 2.Cornell UniversityUSA

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