Summary
The purpose of this note is to derive the asymptotic distributions, means and variances of the Stein estimator, as well as that of the quadratic loss function for the vector case when the population means are nearly equal. These results are given in Section 3 and are obtained by using a method similar to the perturbation method, used by Nagao [4]. In Section 4 exact moments of the Stein estimator are also derived.
Similar content being viewed by others
References
Copson, E. T. (1948).An Introduction to the Theory of Functions of a complex Variable, Oxford University Press, London
Efron, B. and Morris, C. (1972)Simultaneous Estimation of Parameters, The Rand Corporation, 4835.
Efron, B. and Morris, C. (1973). Stein's estimation rule and its competitors—an empirical Bayes approach.J. Amer. Statist. Ass.,68, 117–130.
Nagao, H. (1972). Non-null distributions of the likelihood ratio criteria for independence and equality of mean vectors and covariance matrices.Ann. Inst. Statist. Math.,24, 67–79.
Stein, C. (1962). Confidence sets for the mean of a multivariate normal distribution,J. R. Statist. Soc., B,24, 265–296.
Ullah, A. (1974). On the sampling distribution of improved estimators for coefficients in linear regression.J. Econometrics,2, 143–150.
Van der Merwe, A. J. (1978). The asymptotic expansion of the Stein estimator for the vector case whenθ i ,i=1, ...,p are nearly equal and the distribution of the quadratic loss function ifθ i =0;i=1, ...,p, Technical Report No. 34, Department of Math. Statistics, University of the OFS.
van der Merwe, A. J. (1980). The asymptotic expansion as well as the exact moments of the Stein estimator when the population means are nearly equal,Technical Report No. 55, Department of Math Statistics, University of the OFS.
van der Merwe, A. J. and de Waal, D. J. (1978). The asymptotic expansion of the Stein estimators for the vector case.Ann. Inst. Statist. Math.,30, A, 385–395.
Author information
Authors and Affiliations
Additional information
Financially supported by the CSIR and the University of the OFS Research Fund.
About this article
Cite this article
van der Merwe, A.J. The asymptotic expansion as well as the exact moments of the stein estimator when the population means are nearly equal. Ann Inst Stat Math 35, 31–39 (1983). https://doi.org/10.1007/BF02480961
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02480961