Abstract
A new class of confidence sets for the mean of a p-variate normal distribution (p≥3) is introduced. They are neither spheres nor ellipsoids. We show that we can construct our confidence sets so that their coverage probabilities are equal to the specified confidence coefficient. Some of them are shown to dominate the usual confidence set, a sphere centered at the observations. Numerical results are also given which show how small their volumes are.
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References
Berger, J. O. (1980). A robust generalized Bayes estimator and confidence region for a multivariate normal mean. Ann. Statist., 8, 716–761.
Brown, L. D. (1966). On the admissibility of invariant estimators of one or more location parameters, Ann. Math. Statist., 37, 1087–1136.
Casella, G. and Hwang, J. T. (1983). Empirical Bayes confidence sets for the mean of a multivariate normal distribution, J. Amer. Statist. Assoc., 78, 688–698.
Casella, G. and Hwang, J. T. (1986). Confidence sets and the Stein effect, Comm. Statist. A—Theory Methods, 15, 2043–2063.
Faith, R. E. (1976). Minimax Bayes set and point estimators of a multivariate normal mean, Tech. Report #66, University of Michigan.
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, (supplement issue), Statist. Decisions, 1, 3–16.
Joshi, V. M. (1967). Inadmissibility of the usual confidence sets for the mean of a multivariate normal population, Ann. Math. Statist., 38, 1868–1875.
Shinozaki, N. (1984). Simultaneous estimation of location parameters under quadratic loss, Ann. Statist., 12, 322–335.
Stein, C. (1955). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Proc. Third Berkeley Symp. on Math. Statist. Prob., Vol. 1, 197–206, Univ. of California Press, Berkeley.
Stein, C. (1962). Confidence sets for the mean of a multivariate normal distribution. J. Roy. Statist. Soc. Ser. B, 24, 265–296.
Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist., 9, 1135–1151.
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Shinozaki, N. Improved confidence sets for the mean of a multivariate normal distribution. Ann Inst Stat Math 41, 331–346 (1989). https://doi.org/10.1007/BF00049400
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DOI: https://doi.org/10.1007/BF00049400