Summary
An extension of the method of maximum likelihood leads to a natural solution of the problem raised by Stein, the inadmissibility of the ordinary maximum likelihood estimator for the mean of a multivariate normal distribution.
Similar content being viewed by others
References
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, In2nd International Symposium on Information Theory (B. N. Petrov and F. Csaki, eds.), 267–281, Akademiai Kiado, Budapest.
Akaike, H. (1974). A new look at the statistical model identification,IEEE Trans. Automat. Contr., AC-19, 716–723.
Boltzmann, L. (1877). Uber die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Warmetheorie und der Wahrscheinlichkeitsrechnung respective den Satzen uber das Warmegleichgewicht,Wiener Berichte,76, 373–435.
Dempster, A. P. (1973). Alternatives to least squares in multiple regression, InMultivariate Statistical Inference (D. G. Kabe and R. P. Gupta, eds.), 25–40, North-Holland, Amsterdam.
Efron, B. and Morris, C. (1973). Stein's estimation rule and its competitors—An empirical Bayes approach,J. Amer. Statist. Ass.,68, 117–130.
Efron, B. and Morris, C. (1975). Data analysis using Stein's estimator and its generalizations,J. Amer. Statist. Ass.,70, 311–319.
Good, I. J. (1965).The Estimation of Probabilities, M.I.T. Press, Cambridge.
James, W. and Stein, C. M. (1961). Estimation with quadratic loss function,Proc. 4th Berkeley Symp. Math. Statist. Prob.,1, 361–379.
Kullback, S. (1959).Information and Statistics, Wiley, New York.
Lindley, D. V. (1962). Discussion of a paper by C. Stein,J. R. Statist. Soc., B,24, 285–296.
Sanov, I. N. (1961). On the probability of large deviations of random variables,IMS and AMS Selected Translation in Mathematical Statistics and Probability,1, 213–244.
Stein, C. M. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution,Proc. 3rd Berkeley Symp. Math. Statist. Prob.,1, 197–206.
Stein, C. M. (1962). Confidence sets for the mean of a multivariate normal distribution,J. R. Statist. Soc., B,24, 265–285.
Stein, C. M. (1966). An approach to the recovery of inter-block information in balanced incomplete block designs, InFestschrift for J. Neyman: Research Papers in Statistics (F. N. David, ed.), 351–366, Wiley, New York.
Vincze, I. (1974). On the maximum probability principle in statistical physics, InProgress in Statistics,2, (J. Gani, K. Sarkadi and I. Vince, eds.), 869–893, North-Holland, Amsterdam.
Watson, G. N. (1922).A Treatise on the Theory of Bessel Functions, Cambridge University Press.
Additional information
The Institute of Statistical Mathematics
About this article
Cite this article
Akaike, H. An extension of the method of maximum likelihood and the Stein's problem. Ann Inst Stat Math 29, 153–164 (1977). https://doi.org/10.1007/BF02532781
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02532781