Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-12-02T21:10:53.684Z Has data issue: false hasContentIssue false
  • English
  • Français

THREE RANK FORMULAS ASSOCIATED WITH THE COVARIANCE MATRICES OF THE BLUE AND THE OLSE IN THE GENERAL LINEAR MODEL

Published online by Cambridge University Press:  22 April 2005

Simo Puntanen ,
George P.H. Styan  and
Yongge Tian
Simo Puntanen
Affiliation:
University of Tampere
George P.H. Styan
Affiliation:
McGill University
Yongge Tian
Affiliation:
University of Alberta
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the estimation of the expectation vector Xβ under the general linear model {y,Xβ,σ2V}. We introduce a new handy representation for the rank of the difference of the covariance matrices of the ordinary least squares estimator OLSE(Xβ) (= Hy, say) and the best linear unbiased estimator BLUE(Xβ) (= Gy, say). From this formula some well-known conditions for the equality between Hy and Gy follow at once. We recall that the equality between Hy and Gy can be characterized by the rank-subtractivity ordering between the covariance matrices of y and Hy. This rank characterization suggests a particular presentation for the rank of the difference of the covariance matrices of Hy and Gy. We show, however, that this presentation is valid if and only if the model is connected.

Type
NOTES AND PROBLEMS
Information
Econometric Theory , Volume 21 , Issue 3 , June 2005 , pp. 659 - 663
Copyright
© 2005 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Baksalary, J.K. & S. Puntanen (1990) Characterizations of the best linear unbiased estimator in the general Gauss–Markov model with the use of matrix partial orderings. Linear Algebra and Its Applications 127, 363–370.Google Scholar
Baksalary, J.K. & G.P.H. Styan (1993) Around a formula for the rank of a matrix product with some statistical applications. In R.S. Rees (ed.), Graphs, Matrices, and Designs: Festschrift in Honor of Norman J. Pullman on His Sixtieth Birthday, Lecture Notes in Pure and Applied Mathematics 139, pp. 1–18. Marcel Dekker.
Bérubé, J., R.E. Hartwig, & G.P.H. Styan (1993) On canonical correlations and the degrees of non-orthogonality in the three-way layout. In Kameo Matusita, Madan L. Puri, & Takesi Hayakawa (eds.), Statistical Sciences and Data Analysis: Proceedings of the Third Pacific Area Statistical Conference, pp. 247–252. VSP International Science Publishers BV.
Marsaglia, G. & G.P.H. Styan (1974) Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra 2, 269–292.Google Scholar
Puntanen, S. (1996) Some matrix results related to a partitioned singular linear model. Communications in Statistics: Theory and Methods 25, 269–279.Google Scholar
Puntanen, S. & G.P.H. Styan (1989) The equality of the ordinary least squares estimator and the best linear unbiased estimator (with comments by O. Kempthorne and by S.R. Searle and with “Reply” by the authors). American Statistician 43, 153–164.Google Scholar
Puntanen, S., G.P.H. Styan, & H.J. Werner (2001) Determinant of a skew-symmetric matrix: Solution 1 to Problem 00.1.1 (proposed by Steve Lawford). Econometric Theory 17, 277.
Tian, Y. & G.P.H. Styan (2001) Some rank equalities for idempotent and involutory matrices. Linear Algebra and Its Applications 335, 101–117.Google Scholar
Zyskind, G. & F.B. Martin (1969) On best linear estimation and a general Gauss–Markov theorem in linear models with arbitrary nonnegative covariance structure. SIAM Journal on Applied Mathematics 17, 1190–1202.Google Scholar