Abstract
Based on a sample of size n, we investigate a class of estimators of the mean θ of a p-variate normal distribution with independent components having unknown covariance. This class includes the James-Stein estimator and Lindley's estimator as special cases and was proposed by Stein. The mean squares error improves on that of the sample mean for p≥3. Simple approximations imations for this improvement are given for large n or p. Lindley's estimator improves on that of James and Stein if either n is large, and the “coefficient of variation” of θ is less than a certain increasing function of p, or if p is large. An adaptive estimator is given which for large samples always performs at least as well as these two estimators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions, National Bureau of Standards, Washington, D. C.
Casella, G. and Hwang, J. T. (1982). Limit expressions for the risk of James-Stein estimators, Canad. J. Statist., 10, 305–309.
Copas, J. B. (1983). Regression, prediction and shrinkage, J. Roy. Statist. Soc. Ser B, 45, 311–354.
Efron, B. and Morris, C. (1973a). Combining possibly related estimation problems, J. Roy. Statist. Soc. Ser. B, 35, 379–421.
Efron, B. and Morris, C. (1973b). Stein's estimation rule and its competitors—an empirical Bayes approach, J. Amer. Statist. Assoc., 68, 117–130.
Egerton, M. F. and Laycock, P. J. (1982). An explicit formula for the risk of James-Stein estimators, Canad. J. Statist., 10, 199–205.
George, E. (1986a). Minimax multiple shrinkage estimation, Ann. Statist., 14, 188–205.
George, E. (1986b). A formal Bayes multiple shrinkage estimator, Comm. Statist. A—Theory Methods, 15, 2099–2114.
George, E. (1986c). Combining minimax shrinkage estimators, J. Amer. Statist. Assoc., 81, 437–445.
James, W. and Stein, C. (1961). Estimation with quadratic loss, Proc. Fourth Berkeley Symp. on Math. Statist. Prob., Vol. 1, 361–380, Univ. California Press, Bekeley.
Sathe, Y. S. and Shenoy, R. G. (1986). On sharper bounds for the risk of James-Stein estimators, Comm. Statist. A—Theory Methods, 15, 613–617.
Sclove, S. L., Morris, C. and Radhakrishnan, R. (1972). Nonoptimality of preliminary-test estimators for the mean of a multivariate normal distribution, Ann. Math. Statist., 43, 1481–1490.
Stein, C. (1962). Confidence sets for the mean of a multivariate normal distribution, J. Roy. Statist. Soc. Ser. B, 24, 265–296.
Stein, C. (1966). An approach to the recovery of inter-block information in balanced incomplete block designs, Festschrift for J. Neyman (ed. F. N. David), 351–366, Wiley, New York.
Author information
Authors and Affiliations
About this article
Cite this article
Withers, C.S. A class of multiple shrinkage estimators. Ann Inst Stat Math 43, 147–156 (1991). https://doi.org/10.1007/BF00116474
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00116474