Abstract
The area of decision-theoretic multiparameter estimation in multivariate statistics has been one of intense activity and wide interest over the past few years. Many classical procedures revolve around the eigen structures of random and parameter matrices. Invariance and other considerations tend to focus a great deal of attention on the eigenvalues. The purpose of this paper is to review some of the work relating to eigenvalue estimation.
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References
Anderson, G,A. (1965). ‘An asymptotic expansion for the distribution of the latent roots of the estimated covariance matrix’. Ann, Math. Statist., 36, 1153–1173.
Dey, D,K. and Srinivasan, C, (1984). ‘Estimation of a covariance matrix under Stein’s loss’. Technical Report, Texas Tech, University.
Dey, D,K. and Srinivasan, C. (1985). ‘Estimation of a covariance matrix under Stein’s loss’. Ann. Statist., 13, 1581–1591.
Eaton, M.L. (1970). ‘Some problems in covariance matrix estimation’. Technical Report No. 49, Department of Statistics, Stanford University.
Efron, B. and Morris, C. (1976). ‘Multivariate empirical Bayes estimation of covariance matrices’. Ann. Statist. 4, 22–32.
Haff, L.R. (1977). ‘Minimax estimators for a multinormal precision matrix’. J. Multivariate Anal., 7, 374–385.
Haff, L.R. (1979a). ‘Estimation of the inverse covariance matrix: Random mixtures of the inverse Wishart matrix and the identity’. Ann. Statist., 7, 1264–1276.
Haff, L.R. (1979b). ‘An identity for the Wishart matrix with applications’. J. Multivariate Anal., 9, 531–5421
Haff, L.R. (1980). ‘Empirical Bayes estimation of the multivariate normal covariance matrix’. Ann. Statist., 8, 586–597.
Haff, L.R. (1982). ‘Solutions of the Kuler-Lagrange equations for certain multivariate normal estimation problems’. Unpublished manuscript.
James, W. and Stein, C. (1961), ‘Estimation with quadratic loss’, Proc. Fourth Berkeley Symp, Math. Statist. Prob., 1, 361–379. University of California Press, Berkeley, CA.
Lin, S.F. and Perlman, M.D, (1985). ‘A Monte Carlo comparison of four estimators for a covariance matrix’. In Multivariate Analysis VI (P.R. Krishnaiah, ed,), 411429. North Holland, Amsterdam
Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
Muirhead, R.J. (1985). ‘Estimating a particular function of the multiple correlation coefficient’. J. Amer. Statist. Assoc., 80, 923–925.
Muirhead, R.J. and Leung, P.L. (1985). ‘Estimating functions of canonical correlation coefficients’. Linear Algebra and its Applications, 70, 173–183.
Muirhead, R.J. and Leung, P.L. (1986). ‘Estimation of parameter matrices and eigenvalues in MANOVA and canonical correlation analysis’. Unpublished manuscript.
Muirhead, R.J. and Verathaworni T. (1985), ‘On estimating the latent roots of E1E2 ’. In Multivariate Analysis VI ( P,R. Krishnaiah, ed.), 431–477, North Holland, Amsterdam.
kin, I. and Selliah, J.B. (1977). ‘Estimating covariances in a multivariate normal distribution’, In Statistical Decision Theory and Related Topics II (S.S, Gupta and D. Moore, eds,), 313–326. Academic Press, New York.
Perlman, M.D. and Rasmussen, V.A. (1975). ‘Some remarks on estimating a noncentrality parameter’. Comm. Statist., 4, 455–468.
Stein, C. (1975). ‘Estimation of a covariance matrix’. Rietz Lecture, 39th annual meeting IMS. Atlanta, GA.
Stein, C. (1977a). ‘Estimation of the parameters of a multivariate normal distribution’, Unpublished notes.
Stein, C. (1977b). ‘Lectures on the theory of estimation of many parameters’. (In Russian.) In Studies in the Statistical Theory of Estimation, Part I (I.A. Ibragimol and M.S, Nikulin, eds.), Proceedings of Scientific Seminars of the Steklov Institute, Leningrad Devision, 74, 4–65
Takemura, A. (1984). ‘An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal population’. Tsukuba J. Math, 8, 367–376.
Verathaworn, T. (1983). Decision Theoretic Approaches to Estimating Covariance Matrices in the Multivariate Norme One-Sample Problems. Ph.D. Dissertation, Department of Statistics, University of Michigan.
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Muirhead, R.J. (1987). Developments in Eigenvalue Estimation. In: Gupta, A.K. (eds) Advances in Multivariate Statistical Analysis. Theory and Decision Library, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0653-7_14
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DOI: https://doi.org/10.1007/978-94-017-0653-7_14
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