Elliptically Contoured Distribution (original) (raw)
Abstract
This paper derives an extended version of the Haff or, more appropri-Ž. ately, Stein᎐Haff identity 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.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.
References (18)
- BILODEAU, M. and KARIYA, T. 1989 . Minimax estimators in the normal MANOVA model. J. Multivariate Anal. 28 260᎐270.
- 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. Ž .
- DEY, D. K. AND SRINIVASAN, C. 1985 . Estimation of covariance matrix under Stein's loss. Ann. Statist. 13 1581᎐1591.
- HAFF, L. R. 1979 . An identity for the Wishart distribution with applications. J. Multivariate Anal. 9 531᎐544.
- 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. Ž .
- KUBOKAWA, T. 1998 . The Stein phenomenon in simultaneous estimation: A review. In Applied Ž . Statistical Science 3 S. E. Ahmed, M. Ahsanullah and B. K. Sinha, eds. 143᎐173. NOVA, New York. Ž .
- KUBOKAWA, T. and SRIVASTAVA, M. S. 1997 . Robust improvements in estimation of mean and covariance matrices in elliptically contoured distribution. Discussion Paper Ser. 97-F- 23, Faculty of Economics, Univ. Tokyo.
- ROBERT, C. P. 1994 . The Bayesian Choice: A Decision-Theoretic Motivation. Springer, New York. Ž .
- SHEENA, Y. and TAKEMURA, A. 1992 . Inadmissibility of non-order-preserving orthogonally in- variant estimators of the covariance matrix in the case of Stein's loss. J. Multivariate Anal. 41 117᎐131.
- SRIVASTAVA, M. S. and BILODEAU, M. 1989 . Stein estimation under elliptical distributions. J. Multivariate Anal. 28 247᎐259.
- 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.
- STEIN, C. 1975 . Estimation of a covariance matrix. Rietz Lecture, 39th IMS Annual Meeting, Atlanta, Georgia.
- STEIN, C. 1977a . Estimating the covariance matrix. Unpublished manuscript. Ž .
- STEIN, C. 1977b . Lectures on the theory of estimation of many parameters. In Studies in the Ž . Statistical Theory of Estimation I I. A. Ibragimov and M. S. Nikulin, eds. . Proceed- Ž ings of Scientific Seminars of the Steklov Institute, Leningrad Division 74 4᎐65. In . Russian. Ž .
- SUGIURA, N. and ISHIBAYASHI, H. 1997 . Reference prior Bayes estimator for bivariate nor- mal covariance matrix with risk comparison. Comm. Statist. Theory Methods 26 2203᎐2221. Ž .
- TAKEMURA, A. 1984 . An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal population. Tsukuba J. Math. 8 367᎐376. Ž .
- TAKEMURA, A. 1991 . Foundation of the Multivariate Statistical Inference. Kyoritsu Press, Ž . Tokyo. In Japanese. Ž .
- YANG, R. and BERGER, J. O. 1994 . Estimation of a covariance matrix using the reference prior. Ann. Statist. 22 1195᎐1211.