Abstract
Let F pxp have the multivariate F-distribution with a scale matrix Δ and degrees of freedom n 1and n 2. In this paper the problem of estimating eigenvalues of Δ is considered. By constructing the improved orthogonally invariant estimators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqqHuoaraSqabeaacaqGEbaaaOGaaiikaiaadAeacaGG% Paaaaa!402A!\[\mathop \Delta \limits^{\rm{\^}} (F)\] of Δ, which are analogous to Haff-type estimators of a normal covariance matrix, new estimators of eigenvalues of Δ are given. This is because the eigenvalues of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqqHuoaraSqabeaacaqGEbaaaOGaaiikaiaadAeacaGG% Paaaaa!402A!\[\mathop \Delta \limits^{\rm{\^}} (F)\] are taken as estimates of the eigenvalues of Δ.
Similar content being viewed by others
References
Dey, D. K. (1988). Simultaneous estimation of eigenvalues, Ann. Inst. Statist. Math., 40, 137–147.
Dey, D. K. (1989). On estimation of the scale matrix of the multivariate F distribution, Comm. Statist. A—Theory Methods, 18, 1373–1383.
Gupta, A. K. and Krishnamoorthy, K. (1987). Improved estimators of eigenvalues of 164–1, Tech. Report No. 87–11, Bowling Green State University, Ohio.
Haff, L. R. (1979). An identity for the Wishart distribution with applications, J. Multivariate Anal., 9, 531–544.
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 Solutions of the Euler-Lagrange equations for certain multivariate normal estimation problem (unpublished manuscript).
Konno, Y. (1988). Exact moments of the multivariate F and beta distributions, J. Japan Statist. Soc., 18, 123–130.
Konno, Y. (1991). On estimating eigenvalues of the scale matrix of the multivariate F distribution, Sankhyà Ser. A, 53 (to appear).
Leung, P. L. and Muirhead, R. J. (1987). Estimation of parameter matrices and eigenvalues in MANOVA and canonical correlation analysis, Ann. Statist., 15, 1651–1666.
Leung, P. L. and Muirhead, R. J. (1988). On estimating characteristic roots in a two-sample problem, Statistical Decision Theory and Related Topics IV (eds. S. S. Gupta and J. O. Berger), 397–402, Springer, New York.
Loh, W.-L. (1988). Estimating covariance matrices, Ph. D. Thesis, Stanford University.
Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, Wiley, New York.
Muirhead, R. J. and Verathaworn, T. (1985). On estimating the latent roots of 165–1, Multivariate Analysis—VI (ed. P. R. Krishnaiah), 431–447, North-Holland, Amsterdam.
Perron, F. (1989). Minimax estimators of a covariance matrix, Rapports de recherche du départment de mathématiques et de statistiques, D.M.S. No. 89-23, Université de Montréal.
Author information
Authors and Affiliations
About this article
Cite this article
Konno, Y. A note on estimating eigenvalues of scale matrix of the multivariate F-distribution. Ann Inst Stat Math 43, 157–165 (1991). https://doi.org/10.1007/BF00116475
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00116475