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.
Research performed while visiting the Mathematical Sciences Institute at Cornell University.
Research supported by National Science Foundation Grant No. DMS89-0039. 351
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berger, J. 0. (1985x). Statistical Decision Theory and Bayesian Analysis; Springer (2nd edition).
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.
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.
Berger, J. O. (1990). On the inadmissibility of unbiased estimators. Stat. Prob. Letters 9 (5), 381–384.
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.
Berger, J. O. and Sellke, T. (1987). Testing a point null hypothesis: the irreconcilability of p-values and evidence. JASA 82, 112–122.
Berger, J. O. and Wolpert, R. (1989). The Likelihood Principle. IMS Monograph, Series 6, 2nd edition.
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.
Brown, L. D. (1967). The conditional level of Student’s test. Ann. Statist. 38, 1068–1071.
Brown, L. D. (1975). Estimation with incompletely specified loss functions, J. Amer. Statist. Assoc. 70, 417–26.
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.
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.
George, E. I. and Casella, G. (1990). Empirical Bayes confidence estimation. Technical Report BU-1062-M, Cornell University, Ithaca, NY.
Efron, B. and Morris, C. (1973). Stein estimation and its competitor - an Empirical Bayes approach. JASA 68, 117–130.
Hwang, J. T. and Brown, L. D. (1991). The estimated confidence approach under the validity constraint criterion. Ann. Stat. 19, 1964–1977.
Hwang, J. T. and Casella, G. (1982). Minimax confidence sets for the mean of a multivariate normal distribution. Ann. Stat. 10, 868–881.
Hwang, J. T., Casella, G., Robert, C., Wells, M. T., Farrell, R. (1992). Estimation of accuracy in testing. Ann. Stat. 20, 490–509.
Hwang, J. T. and Ullah, A. (1989). Confidence seta centered at James-Stein estimators. Technical Report, Mathematics Department, Cornell University.
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.
Kass, R. and Steffey, D. (1989). Parametric Empirical Bayes Models. JASA 86, 717–726.
Lu, K. and Berger, J. (1989x). Estimated confidence procedures for multivariate normal means. J. 5tat. Plans. Inf. 23, 1–19.
Lu, K. and Berger, J. (1989b). Estimation of normal means: Frequentists estimators of loss. Ann. Stat. 17, 890–907.
Morris, C. (1983). Parametric Empirical Bayes Inference: Theory and Applications. JASA 78, 47–65.
Robert, C. and Casella, G. (1990). Improved confidence sets for spherically symmetric distributions. J. Multivariate Analysis 32 (1), 84–94.
Robinson, G. (1979a). Conditional properties of statistical procedures. Ann. Stat. 7, 742–755.
Robinson, G. K. (1979b). Conditional properties of statistical procedures for location and scale parameters. Ann. Stat. 7, 756–771.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Robert, C., Casella, G. (1994). Improved Confidence Statements for the Usual Multivariate Normal Confidence Set. In: Gupta, S.S., Berger, J.O. (eds) Statistical Decision Theory and Related Topics V. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2618-5_26
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2618-5_26
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7609-8
Online ISBN: 978-1-4612-2618-5
eBook Packages: Springer Book Archive