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Robust improvement in estimation of a covariance matrix in an elliptically contoured distribution. (English) Zbl 0942.62063
Summary: This paper derives an extended version of the Haff or, more appropriately, Stein-Haff identity [C. Stein, Estimating the covariance matrix. Unpublished manuscript. (1977); L. R. Haff, J. Multivariate Anal. 9, 531-544 (1979; Zbl 0423.62036)] for an elliptically contoured distribution (ECD). This identity is then used to show that the minimax estimators of the covariance matrix obtained under normal models remain robust under the ECD model.

MSC:
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
62C99 Statistical decision theory
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[1] BILODEAU, M. and KARIYA, T. 1989. Minimax estimators in the normal MANOVA model. J. Multivariate Anal. 28 260 270.Z. · Zbl 0683.62033
[2] CELLIER, D., FOURDRINIER, D. AND ROBERT, C. 1989. Robust shrinkage estimators of the location parameter for elliptically symmetric distributions. J. Multivariate Anal. 29 39 52. Z. · Zbl 0678.62061
[3] DEY, D. K. AND SRINIVASAN, C. 1985. Estimation of covariance matrix under Stein’s loss. Ann. Statist. 13 1581 1591. Z. · Zbl 0582.62042
[4] HAFF, L. R. 1979. An identity for the Wishart distribution with applications. J. Multivariate Anal. 9 531 544. Z. · Zbl 0423.62036
[5] JAMES, W. and STEIN, C. 1961. Estimation with quadratic loss. In Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361 379. Univ. California Press, Berkeley. Z. · Zbl 1281.62026
[6] KUBOKAWA, T. 1998. The Stein phenomenon in simultaneous estimation: A review. In Applied Z. Statistical Science 3 S. E. Ahmed, M. Ahsanullah and B. K. Sinha, eds. 143 173.
[7] NOVA, New York. Z.
[8] KUBOKAWA, T. and SRIVASTAVA, M. S. 1997. Robust improvements in estimation of mean and covariance matrices in elliptically contoured distribution. Discussion Paper Ser. 97-F23, Faculty of Economics, Univ. Tokyo.
[9] ROBERT, C. P. 1994. The Bayesian Choice: A Decision-Theoretic Motivation. Springer, New York. Z. · Zbl 0808.62005
[10] SHEENA, Y. and TAKEMURA, A. 1992. Inadmissibility of non-order-preserving orthogonally invariant estimators of the covariance matrix in the case of Stein’s loss. J. Multivariate Anal. 41 117 131. Z. · Zbl 0764.62006
[11] SRIVASTAVA, M. S. and BILODEAU, M. 1989. Stein estimation under elliptical distributions. J. Multivariate Anal. 28 247 259. Z. · Zbl 0667.62039
[12] STEIN, C. 1956. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proc. Third Berkeley Symp. Math. Statist. Probab. 1 197 206. Univ. California Press, Berkeley. Z. · Zbl 0073.35602
[13] STEIN, C. 1975. Estimation of a covariance matrix. Rietz Lecture, 39th IMS Annual Meeting, Atlanta, Georgia. Z.
[14] STEIN, C. 1977a. Estimating the covariance matrix. Unpublished manuscript. Z. JSTOR: · Zbl 0992.11068
[15] STEIN, C. 1977b. Lectures on the theory of estimation of many parameters. In Studies in the Z. Statistical Theory of Estimation I I. A. Ibragimov and M. S. Nikulin, eds.. ProceedZ ings of Scientific Seminars of the Steklov Institute, Leningrad Division 74 4 65. In. Russian. Z. JSTOR: · Zbl 0992.11068
[16] SUGIURA, N. and ISHIBAYASHI, H. 1997. Reference prior Bayes estimator for bivariate normal covariance matrix with risk comparison. Comm. Statist. Theory Methods 26 2203 2221. Z. · Zbl 0954.62539
[17] TAKEMURA, A. 1984. An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal population. Tsukuba J. Math. 8 367 376. Z. · Zbl 0565.62035
[18] TAKEMURA, A. 1991. Foundation of the Multivariate Statistical Inference. Kyoritsu Press, Z. Tokyo. In Japanese. Z.
[19] YANG, R. and BERGER, J. O. 1994. Estimation of a covariance matrix using the reference prior. Ann. Statist. 22 1195 1211. · Zbl 0819.62013
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