Abstract
Estimation problems which are invariant under an amenable group are considered, under regularity conditions and the assumption that a unique minimum risk equivariant estimator δ E exists. We show that for any shrinkage estimator, δ S , whose risk is everywhere better than that of δ E , δ S (x) is close to δ S (x) except on a set of x-values which is small in a sense which we make precise. Further, the values of the risk functions of δ E and δ S are close except on a small set.
As a corollary we show that when independent estimation problems are combined, large Stein effects occur only on small sets. In Appendix I, we show that a similar result holds if the parameter space is compact, whether the model is invariant or not.
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
Bondar, J., and P. Milnes (1981), “Amenability: a survey for statistical applications of Hunt-Stein and related conditions on groups.”Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 57, 103– 128.
Brown, L. D. (1966), “On the admissibility of invariant estimators of one or more location parameters.”Annals of Mathematical Statistics 37, 1087– 1136.
Gutmann, S. (1982), “Stein’s paradox is impossible with finite sample space.”Annals of Statistics 10, 1017– 1020.
Heath, D., and W. Sudderth (1978), “On finitely additive priors, coherence, and extended admissibility.”Annals of Statistics 6, 333– 345.
Hwang, J. T., and G. Casella (1982), “Minimax confidence sets for the mean of a multivariate normal distribution.”Annals of Statistics 10, 868– 881.
Joshi, V. M. (1967), “Inadmissibility of the usual confidence sets for the mean of a multivariate normal distribution.”Annals of Mathematical Statistics 38, 1868– 1875.
Joshi, V. M. (1979), “Joint admissibility of the sample means as estimators of the means of finite populations.”Annals of Statistics 7, 995– 1002.
Lehmann, E. (1959),Testing Statistical Hypotheses. New York: Wiley and Sons.
Lehmann, E. (1983),Theory of Point Estimation. New York: Wiley and Sons.
Stone, M. (1970), “Necessary and sufficient condition for convergence in probability to invariant posterior distributions.”Annals of Mathematical Statistics 41, 1349– 1353.
Wald, A. (1950),Statistical Decision Functions. Reprinted by Chelsea.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Bondar, J.V. (1987). How Much Improvement can a Shrinkage Estimator Give?. In: MacNeill, I.B., Umphrey, G.J., Safiul Haq, M., Harper, W.L., Provost, S.B. (eds) Advances in the Statistical Sciences: Foundations of Statistical Inference. The University of Western Ontario Series in Philosophy of Science, vol 35. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4788-7_9
Download citation
DOI: https://doi.org/10.1007/978-94-009-4788-7_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8623-3
Online ISBN: 978-94-009-4788-7
eBook Packages: Springer Book Archive