Abstract
The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of 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.
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References
Ahmed SE (1992) Large-sample pooling procedure for correlation. The Statisticial 40:425–438
Ahmed SE, Saleh AKME (1993) Improved estimation for the component meanvector. Japan Journal of Statistics 43:177–195
Anderson TW (1963) Asymptotic theory for principal component analysis, Annals of Mathematical Statistics 34:122–148
Berger JO (1985) Statistical decision theory and Bayesian analysis, second edition. Springer-Verlag, New York
Brandwein AC, Strawderman WE (1990) Stein estimation: The spherically symmetric case. Statistical Science 5:356–369
Dey DK (1988) Simultaneous estimation of eigenvalues. Ann Inst Statist Math 40:137–147
Dey DK, Srinivasan C (1986) Trimmed minimax estimator of a covariance matrix. Ann Inst Statist Math 38:47–54
Grishick MA (1939) On the sampling theory of roots of determinantal equations. Annals of Mathematical Statistics 10:203–224
Hoffmann K (1992) Improved estimation of distribution parameters: Stein-Type estimators. B. G. Teubner Verlgsgesellschaft, Stuttgart
Joarder AH, Ahmed SE (1996) Estimation of the characteristic roots of the scale matrix. Metrika 44:259–267
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
Judge GG, Bock ME (1978) The statistical implication of pre-test and Stein-rule estimators in econometrics. North-Holland, Amsterdam
Leung PL (1992) Estimation of eigenvalues of the scale matrix of the multivariate F distribution. Communications in Statistics. Theory and methods 21:1845–1856
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
Robert CP (1994) The Bayesian choice: A decision-theoritic motivation. Springer-Verlag, New York
Rukhin AL (1995) Admissibility: Survey of concept in progress. International Statistical Review 63:95–115
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
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
Stigler SM (1990) The 1988 Neyman Memorial Lecture: A Galtonian perspective on shrinkage estimators. Statistical Science 5:147–155
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Ahmed, S.E. Large-sample estimation strategies for eigenvalues of a Wishart matrix. Metrika 47, 35–45 (1998). https://doi.org/10.1007/BF02742863
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DOI: https://doi.org/10.1007/BF02742863