Abstract
In this paper, the problems of estimating the covariance matrix in a Wishart distribution (refer as one-sample problem) and the scale matrix in a multi-variate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their harmonic mean is proposed. It is shown that the new estimator dominates the best linear estimator under two scale invariant loss functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dey, D. K. (1988). Simultaneous estimation of eigenvalues, Ann. Inst. Statist. Math., 40, 137-147.
Dey, D. K. (1989). On estimation of the scale matrix of the multivariate F distribution, Comm. Statist. Theory Methods, 18, 1373-1383.
Dey, D. K. and Srinivasan, C. (1985). Estimation of a covariance matrix under Stein's loss, Ann. Statist., 13, 1581-1591.
Friedman, J. H. (1989). Regularized discriminant analysis, J. Amer. Statist. Assoc., 84, 165-175.
Gupta, A. K. and Krishnamoorthy, K. (1990). Improved estimators of eigenvalues of Σ1Σ −12 , Statist. Decisions, 8, 247-263.
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.
Konno, Y. (1988). Exact moments of the multivariate F and beta distributions, J. Japan Statist. Soc., 18, 123-130.
Konno, Y. (1991). A note on estimating eigenvalues of scale matrix of the multivariate F-distribution, Ann. Inst. Statist. Math., 43, 157-165.
Leung, P. L. (1992). Estimation of eigenvalues of the scale matrix of the multivariate F distribution, Comm. Statist. Theory Methods, 21, 1845-1856.
Leung, P. L. and Chan, W. Y. (1998). Estimation of the scale matrix and its eigenvalues in the Wishart and the Multivariate F distributions, Ann. Inst. Statist. Math., 50, 523-530.
Leung, P. L., and Muirhead, R. J. (1988). On estimating characteristic roots in a two—sample problem, Statistical Decision Theory and Related Topics IV, Vol. 1 (eds. S. S. Gupta and J. O. Berger), 397-402, Springer, New York.
Lin, S. P. and Perlman, M. D. (1985). A Monte Carlo comparison of four estimators for a covariance matrix, Multivariate Analysis VI (ed. P. R. Krishnaiah), 411-429, North Holland, Amsterdam.
Merritt, F. S. (1962). Mathematics Manual, McGraw-Hill. New York.
Muirhead, R. J. (1987). Developments in eigenvalue estimation, Advances in Multivariate Statistical Analysis, 277-288, Reidel, Dordrecht.
Muirhead, R. J. and Verathaworn, T. (1985). On estimating the latent roots of Σ1Σ −12 , Multivariate Analysis VI (ed. P. R. Krishnaiah), 431-447, North-Holland, Amsterdam.
Pal, N. (1993). Estimating the normal dispersion matrix and the precision matrix from a decision-theoretic point of view: A review, Statist. Papers, 34, 1-26.
Sheena, Y. and Takemura, A. (1992). Inadmissibility of non-order-preserving othogonally invariant estimators of the covariance matrix in the case of Stein's loss, J. Multivariate Anal., 41, 117-131.
Stein, C. (1975). Estimation of a covariance matrix, Rietz Lecture, 39th Annual meeting IMS, Atlanta, Georgia.
Author information
Authors and Affiliations
About this article
Cite this article
Leung, P.L., Ng, F.Y. Improved Estimation of Parameter Matrices in a One-Sample and Two-Sample Problems. Annals of the Institute of Statistical Mathematics 53, 769–780 (2001). https://doi.org/10.1023/A:1014661221279
Issue Date:
DOI: https://doi.org/10.1023/A:1014661221279