ArticleOriginal scientific text

Title

Inverting covariance matrices

Authors 1, 2

Affiliations

  1. Institute of Mathematics, University of Rzeszów, Rejtana 16 A, P.O. Box 155, 35-959 Rzeszów, Poland
  2. Department of Statistics and Econometrics, Maria Curie-Skłodowska University, Pl. Marii Curie Skłodowskiej 5, 20-031 Lublin, Poland

Abstract

Some useful tools in modelling linear experiments with general multi-way classification of the random effects and some convenient forms of the covariance matrix and its inverse are presented. Moreover, the Sherman-Morrison-Woodbury formula is applied for inverting the covariance matrix in such experiments.

Keywords

multi-way classification, cross, hierarchical, balanced, unbalanced, covariance matrix, inverting

Bibliography

  1. G.H. Golub and C.F. Van Loan, Matrix Computation, Sec. Edition, J. Hopkins Univ. Press, Baltimore 1989.
  2. F.A. Graybill, Matrices with Application in Statistics, Sec. Edition, Wadsworth, Belmont, CA 1983.
  3. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge 1985.
  4. J. Jiang, Dispersion matrix in balanced mixed ANOVA models, Linear Algebra Appl. 382 (2004), 211-219.
  5. J. Kleffe and B. Seifert, Computation of variance components by MINQUE method, J. Multivariate Anal. 18 (1986), 107-116.
  6. L.R. LaMotte, Notes on the covariance matrix of a random nested ANOVA model, Ann. Math. Statist. 43 (1972), 659-662.
  7. C.R. Rao, Linear Statistical Inference and Its Applications, Sec. Edition, J. Wiley, New York 1973.
  8. S.R. Searle, G. Casella and C. McCulloch, Variance Components, J. Wiley, New York 1992.
  9. J. Seely, Quadratic subspaces and completeness, Ann. Math. Statist. 42 (1971), 710-721.
  10. C. Stępniak, A note on estimation of parameters in linear models, Bull. Acad. Polon. Sc. Math., Astr. et Phys. 22 (1974), 1151-1154.
  11. C. Stępniak, Optimal allocation of units in experimental designs with hierarchical and cross classification, Ann. Inst. Statist. Math. A 35 (1983), 461-473.
  12. C. Stępniak, Inversion of covariance matrices: explicit formulae, SIAM J. Matrix Anal. Appl. 12 (1991), 577-580.
  13. C. Stępniak and M. Niezgoda, Inverting covariance matrices in unbalanced hierarchical models, J. Statist. Comput. Simul. 51 (1995), 215-221.
  14. D.M. VanLeeuwen, D.S. Birkes and J.F. Seely, Balance and orthogonality in designs for mixed classification models, Ann. Statist. 2 (1999), 1927-1947.
  15. R. Zmyślony and H. Drygas, Jordan algebras and Bayesian quadratic estimation of variance components, Linear Algebra Appl. 168 (1992), 259-275.
Pages:
163-177
Main language of publication
English
Received
2006-05-15
Accepted
2007-01-24
Published
2006
Exact and natural sciences