Abstract
Approaches to set estimation based on a decision-theoretic formulation have usually used a loss function that is a linear combination of volume and coverage probability. Such loss functions can suffer from paradoxical behavior of the Bayes rules, and thus may not be appropriate. We investigate the behavior of optimal set estimators for different classes of loss functions and study their decision-theoretic properties.
Research supported by National Science Foundation Grant No. DMS91-00839.
Research supported by National Science Foundation Grant No. DMS88-09016.
Research supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University.
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. O. (1985). Statistical Decision Theory and Bayesian Analysis. 2nd edition, Springer-Verlag.
Blyth, C. R. and Hutchinson, D. W. (1961). Tables of Neyman-Shortest Confidence Intervals for the Binomial Parameter. Biometrika 47, 381–391.
Casella, G. and Hwang, J. T. (1991). Evaluating confidence sets using loss functions. Statist. Sinica 1, 159–173.
Casella, G., Hwang, J. T. G., and Robert, C. P. (1993). A paradox in decision-theoretic interval estimation. Statist. Sinica 3 (1), 141–155.
Cohen, A. and Strawderman, W. E. (1973). Admissibility implications for different criteria in confidence estimation. Ann. Statist. 1, 363–366.
DeGroot, M. (1970). Optimal Statistical Decisions. John Wiley & Sons, New York.
Hwang, J. T. and Casella, G. (1982). Minimax confidence sets for the mean of a multivariate normal distribution. Ann. Statist. 10, 868–881.
Hwang, J. T. and Casella, G. (1984). Improved set estimators for a multivariate normal mean. Statistics and Decisions, Supplement Issue 1, 3–16.
Joshi, V. M. (1969). Admissibility of the usual confidence set for the mean of a univariate or bivariate normal population. Ann. Math. Statist. 40, 1042–1067.
Lindley, D. (1985). Making Decisions. John Wiley & Sons, New York.
Meeden, G. and Vardeman, S. (1985). Bayes and admissible set estimation. J. Amer. Statist. Assoc. 80, 465–471.
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
Casella, G., Hwang, J.T.G., Robert, C.P. (1994). Loss Functions for Set Estimation. 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_18
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2618-5_18
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