Summary
In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator.
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References
Barlow, R. E. et al. (1972).Statistical Inference Under Order Restrictions, John Wiley and Sons, New York.
Dey, D. K. and Berger, J. (1983). On truncation of shrinkage estimators in simultaneous estimation of normal means,J. Amer. Statist. Ass.,78, 865–869.
Dey, D. K., Ghosh, M. and Srinivasan, C. (1986). Simultaneous estimation of parameters under entropy loss, to appear inJ. Statist. Plann. Inf.
Dey, D. K. and Srinivasan, C. (1985). Estimation of covariance matrix under Stein's loss,Ann. Statist.,13, 1581–1591.
Efron, B. and Morris, C. (1976). Multivariate empirical Bayes estimation of covariance matrices,Ann. Statist.,4, 22–32.
Ghosh, M. and Dey, D. K. (1984). Trimmed estimates in simultaneous estimation of parameters in exponential families,J. Multivariate Anal.,15, 183–200.
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 distribution with applications,J. Multivariate Anal.,9, 531–542.
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 Solutions of the Euler-Lagrange equations for certain multivariate normal estimation problems, unpublished manuscript.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, 361–380, Univ. of California Press.
Lin, S. P. (1977). Improved procedures for estimating a correlation matrix, unpublished Ph.D. thesis, Department of Statistics, University of Chicago.
Lin, S. P. and Perlman, M. D. (1985). A Monte Carlo comparison of four estimators for a covariance matrix,Multivariate Analysis, (ed. P. R. Krishnaiah),6, 411–429, North Holland, Amsterdam.
Stein, C. (1975). Rietz lecture,38th Annual Meeting IMS, Atlanta, Georgia.
Stein, C. (1977). Unpublished notes on estimating the covariance matrix.
Stein, C. (1981). Estimation of the mean of multivariate normal distribution,Ann. Statist.,9, 1135–1151.
Takemura, A. (1984). An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal population,Tsukuba J. Math.,8, 367–376.
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The second author's research was supported by NSF Grant Number MCS 82-12968.
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Dey, D.K., Srinivasan, C. Trimmed minimax estimator of a covariance matrix. Ann Inst Stat Math 38, 101–108 (1986). https://doi.org/10.1007/BF02482503
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DOI: https://doi.org/10.1007/BF02482503