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Large-sample estimation strategies for eigenvalues of a Wishart matrix. (English) Zbl 1093.62543
Summary: The problem of simultaneous asymptotic estimation of eigenvalues of the covariance matrix of a Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.

62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
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[1] Ahmed SE (1992) Large-sample pooling procedure for correlation. The Statisticial 40:425–438 · doi:10.2307/2349007
[2] Ahmed SE, Saleh AKME (1993) Improved estimation for the component meanvector. Japan Journal of Statistics 43:177–195
[3] Anderson TW (1963) Asymptotic theory for principal component analysis, Annals of Mathematical Statistics 34:122–148 · Zbl 0202.49504 · doi:10.1214/aoms/1177704248
[4] Berger JO (1985) Statistical decision theory and Bayesian analysis, second edition. Springer-Verlag, New York
[5] Brandwein AC, Strawderman WE (1990) Stein estimation: The spherically symmetric case. Statistical Science 5:356–369 · Zbl 0955.62611 · doi:10.1214/ss/1177012104
[6] Dey DK (1988) Simultaneous estimation of eigenvalues. Ann Inst Statist Math 40:137–147 · Zbl 0669.62034 · doi:10.1007/BF00053961
[7] Dey DK, Srinivasan C (1986) Trimmed minimax estimator of a covariance matrix. Ann Inst Statist Math 38:47–54 · Zbl 0591.62048 · doi:10.1007/BF02482503
[8] Grishick MA (1939) On the sampling theory of roots of determinantal equations. Annals of Mathematical Statistics 10:203–224 · Zbl 0023.24503 · doi:10.1214/aoms/1177732180
[9] Hoffmann K (1992) Improved estimation of distribution parameters: Stein-Type estimators. B. G. Teubner Verlgsgesellschaft, Stuttgart · Zbl 0762.62003
[10] Joarder AH, Ahmed SE (1996) Estimation of the characteristic roots of the scale matrix. Metrika 44:259–267 · Zbl 0864.62038 · doi:10.1007/BF02614070
[11] James W, Stein C (1961) Estimation with quadratic loss. Proceeding of the fourth Berkeley symposium on Mathematical statistics and Probability, University of California Press, pp. 361–379 · Zbl 1281.62026
[12] Judge GG, Bock ME (1978) The statistical implication of pre-test and Stein-rule estimators in econometrics. North-Holland, Amsterdam · Zbl 0395.62078
[13] Leung PL (1992) Estimation of eigenvalues of the scale matrix of the multivariate F distribution. Communications in Statistics. Theory and methods 21:1845–1856 · Zbl 0775.62133 · doi:10.1080/03610929208830883
[14] Olkin I, Selliah JB (1977) Estimating covariance matrix in a multivariate normal distribution In: Gupta SS, Moore D (eds) Statistical decision theory and related topics, II, Academic Press, New York, pp. 313–326
[15] Robert CP (1994) The Bayesian choice: A decision-theoritic motivation. Springer-Verlag, New York
[16] Rukhin AL (1995) Admissibility: Survey of concept in progress. International Statistical Review 63:95–115 · Zbl 0845.62005 · doi:10.2307/1403779
[17] Sclove SL, Morris C, Radhakrishnan R (1972) Non-optimality of preliminary test estimators of the mean of a multivariate normal distribution. Annals of Mathematical Statistics 43:1481–1490 · Zbl 0249.62029 · doi:10.1214/aoms/1177692380
[18] Stein C (1956) Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proceeding of the third Berkeley symposium on Mathematical statistics and Probability, University of California Press, volume 1, pp. 197–206 · Zbl 0073.35602
[19] Stigler SM (1990) The 1988 Neyman Memorial Lecture: A Galtonian perspective on shrinkage estimators. Statistical Science 5:147–155 · Zbl 0955.62610
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