Abstract
For estimating the power of a generalized variance under a multivariate normal distribution with unknown means, the inadmissibility of the best affine equivariant estimator relative to the symmetric loss is shown, and a class of improved estimators is given. The problem of estimating the covariance matrix is also discussed.
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References
Dey, D. K. and Srinivasan, C. (1985). Estimation of a 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.
Haff, L. R. (1980). Empirical Bayes estimation of the multivariate normal covariance matrix, Ann. Statist., 8, 586–597.
James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist., 35, 475–501.
James, W. and Stein, C. (1961). Estimation with quadratic loss, Proc. Fourth Berkeley Symp. on Math. Statist. Prob., Vol. 1, 361–380, Univ. of California Press.
Muirhead, R. J. (1982) Aspects of Multivariate Statistical Theory, Wiley, New York.
Olkin, I. and Selliah, J. B. (1977). Estimating covariances in a multivariate normal distribution, Statistical Decision Theory and Related Topics, II, (eds. S. S. Gupta and D. S. Moore), 313–326, Academic Press, New York.
Pal, N. (1988). Decision-theoretic estimation of generalized variance and generalized precision, Comm. Statist. A—Theory Methods. 17, 4221–4230.
Shorrock, R. W. and Zidek, J. V. (1976). An improved estimator of the generalized variance, Ann. Statist., 4, 629–638.
Sinha, B. K. (1976). On improved estimators of the generalized variance, J. Multivariate Anal., 6, 617–625.
Sinha, B. K. and Ghosh, M. (1987). Inadmissibility of the best equivariant estimators of the variance-covariance matrix, the precision matrix, and the generalized variance under entropy loss, Statist. Decisions, 5, 201–227.
Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Ann. Inst. Statist. Math., 16, 155–160.
Strawderman, W. E. (1974). Minimax estimation of powers of the variance of a normal population under squared error loss, Ann. Statist., 2, 190–198.
Sugiura, N. (1988). A class of improved estimators of powers of the generalized variance and precision under squared loss, Statistical Theory and Data Analysis II, (ed. K. Matusita), North-Holland, Amsterdam.
Sugiura, N. (1989). Entropy loss and a class of improved estimators for powers of the generalized variance, (to appear in Sankhyā).
Sugiura, N. and Konno, Y. (1987). Risk of improved estimators for generalized variance and precision, Advances in Multivariate Statistical Analysis, (ed. A. K. Gupta), 353–371, Reidel, Dordrecht.
Sugiura, N. and Konno, Y. (1988). Entropy loss and risk of improved estimators for the generalized variance and precision, Ann. Inst. Statist. Math., 40, 329–341.
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Kubokawa, T., Konno, Y. Estimating the covariance matrix and the generalized variance under a symmetric loss. Ann Inst Stat Math 42, 331–343 (1990). https://doi.org/10.1007/BF00050840
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DOI: https://doi.org/10.1007/BF00050840