, Volume 47, Issue 1, pp 35–45 | Cite as

Large-sample estimation strategies for eigenvalues of a Wishart matrix

  • S. E. Ahmed


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.

Key Words and Phrases

Wishart distribution eigenvalues covariance matrix James-Stein type estimators positive-part estimators asymptotic quadratic bias and risk 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahmed SE (1992) Large-sample pooling procedure for correlation. The Statisticial 40:425–438CrossRefGoogle Scholar
  2. Ahmed SE, Saleh AKME (1993) Improved estimation for the component meanvector. Japan Journal of Statistics 43:177–195Google Scholar
  3. Anderson TW (1963) Asymptotic theory for principal component analysis, Annals of Mathematical Statistics 34:122–148MathSciNetMATHGoogle Scholar
  4. Berger JO (1985) Statistical decision theory and Bayesian analysis, second edition. Springer-Verlag, New YorkMATHGoogle Scholar
  5. Brandwein AC, Strawderman WE (1990) Stein estimation: The spherically symmetric case. Statistical Science 5:356–369MATHMathSciNetGoogle Scholar
  6. Dey DK (1988) Simultaneous estimation of eigenvalues. Ann Inst Statist Math 40:137–147MATHCrossRefMathSciNetGoogle Scholar
  7. Dey DK, Srinivasan C (1986) Trimmed minimax estimator of a covariance matrix. Ann Inst Statist Math 38:47–54CrossRefMathSciNetGoogle Scholar
  8. Grishick MA (1939) On the sampling theory of roots of determinantal equations. Annals of Mathematical Statistics 10:203–224MathSciNetGoogle Scholar
  9. Hoffmann K (1992) Improved estimation of distribution parameters: Stein-Type estimators. B. G. Teubner Verlgsgesellschaft, StuttgartMATHGoogle Scholar
  10. Joarder AH, Ahmed SE (1996) Estimation of the characteristic roots of the scale matrix. Metrika 44:259–267MATHCrossRefMathSciNetGoogle Scholar
  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. 361–379Google Scholar
  12. Judge GG, Bock ME (1978) The statistical implication of pre-test and Stein-rule estimators in econometrics. North-Holland, AmsterdamGoogle Scholar
  13. Leung PL (1992) Estimation of eigenvalues of the scale matrix of the multivariate F distribution. Communications in Statistics. Theory and methods 21:1845–1856MATHGoogle Scholar
  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. 313–326Google Scholar
  15. Robert CP (1994) The Bayesian choice: A decision-theoritic motivation. Springer-Verlag, New YorkGoogle Scholar
  16. Rukhin AL (1995) Admissibility: Survey of concept in progress. International Statistical Review 63:95–115MATHCrossRefGoogle Scholar
  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 43:1481–1490MathSciNetMATHGoogle Scholar
  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. 197–206Google Scholar
  19. Stigler SM (1990) The 1988 Neyman Memorial Lecture: A Galtonian perspective on shrinkage estimators. Statistical Science 5:147–155MATHMathSciNetGoogle Scholar

Copyright information

© Physica-Verlag 1998

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Regina, ReginaSaskatchewanCanada

Personalised recommendations