Abstract
When an independent estimate of covariance matrix is available, we often prefer two–stage estimate (TSE). Expressions of exact covariance matrix of the TSE obtained by using all and some covariables in covariance adjustment approach are given, and a necessary and sufficient condition for the TSE to be superior to the least square estimate and related large sample test is also established. Furthermore the TSE, by using some covariables, is expressed as weighted least square estimate. Basing on this fact, a necessary and sufficient condition for the TSE by using some covariables to be superior to the TSE by using all covariables is obtained. These results give us some insight into the selection of covariables in the TSE and its application.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zhang, B. X., Liu, B. S., Lu, C. Y.: A study of the equivalence of the BLUES between a partitioned sigular linear model and its reduced sigular linear models. Acta Mathematica Sinica, English Series, 20(3), 557–568 (2004)
Toyooka, Y., Kariya, T.: An approach to upper bound problems for risks generalized least squares estimator. The Ann. Statist., 14, 679–685 (1986)
Wang, S. G, Liu, A. Y.: The relative e.ciency of the two-stage estimate. Acta Mathematica Sinica, Chinese Series, 32(1), 42–50 (1989)
Wang, S. G, Fan, Y. H.: On properties of the two-stage estimate in Panel models. Chinese Journal of Applied Probability and Statistics, 14(2), 177–184 (1998) (in Chinese)
Rao, C. R.: Least squares theory using an estimated dispersion matrix and its application to measurement of signals, In Proceedings of the Fifth Berkeley Symposium on Math, Statist & Prob. Eds. by Lecam, J and Neyman, J., 1, 355, 1967
Wang, S. G, Yang, Z. H.: The Pitman property of the covariance adjusted estimate. Chinese Science Bulletin, 40, 12–15 (1995) (in Chinese)
Chen, X. R, Wang S. G.: Least Squares in Linear Models, Shanghai Science and Technology Press, 83, 2003 (in Chinese)
Khatri, C. G.: A note on a MANOVA model applied to problems in growth curve. Ann. Inst. Statist. Math., 18, 75–86 (1966)
Muirhead, R. J.: Aspects of Multivariate Statistical Theory, John Wiley, New York, 1982
Hsu, P. L.: On the limiting distribution of the canonical correlations. Biometrika, 32, 38–45 (1941)
Author information
Authors and Affiliations
Corresponding authors
Additional information
The work is supported by the National Natural Science Foundation of China (10271010), the Natural Science Foundation of Beijing (1032001)
Rights and permissions
About this article
Cite this article
Yin, S.J., Wang, S.G. On Two–stage Estimate Based on Independent Estimate of Covariance Matrix. Acta Math Sinica 22, 283–288 (2006). https://doi.org/10.1007/s10114-005-0650-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-005-0650-1