Journal of Statistical Theory and Practice

, Volume 4, Issue 2, pp 233–242 | Cite as

Nesting Segregated Mixed Models

  • S. S. FerreiraEmail author
  • D. Ferreira
  • C. Nunes
  • J. T. Mexia


A mixed model has segregation when its random effects part is segregated as a sub-model. It will be shown that under orthogonality condition, nesting a random effects model inside a segregated mixed model or a segregated mixed model inside a fixed effects model the result will be a segregated mixed model. Unbiased estimators will be obtained for the variance components in both classes of models which are UMVUE, once normality is assumed.

AMS Subject Classification

62K15 62E15 62H10 62H15 62J10 


Normal orthogonal mixed models Segregated mixed model 


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  1. Ferreira, S.S., 2006. Inference for Orthogonal Models with Segregation. PhD Thesis, UBI — Covilhã.Google Scholar
  2. Fonseca, M., Mexia, J.T., Zmyślony, R., 2003. Estimators and tests for variance components in cross nested orthogonal models. Discussiones Mathematicae — Probability and Statistics, 23, 173–201.zbMATHGoogle Scholar
  3. Fonseca, M., Mexia, J.T., Zmyślony, R., 2006. Binary operations on Jordan algebras and orthogonal normal models. Linear Algebra and It’s Applications,417(1), 75–86.MathSciNetCrossRefGoogle Scholar
  4. Fonseca, M., Mexia, J.T., Zmyślony, R., 2007. Pivot variables and orthogonal normal models. Journal of Interdisciplinary Mathematics, 10, 305–326.MathSciNetCrossRefGoogle Scholar
  5. Jordan, P., Von Neumann, J., Wigner, E., 1934. On an algebraic generalization of the quantum mechanical formulation. The Annals of Mathematical Statistics, 35(1).Google Scholar
  6. Khuri A.I., Matthew T., Sinha B.K., 1998. Statistical Tests for Mixed Linear Models. John Wiley & Sons, New York.CrossRefGoogle Scholar
  7. Lehman, E.L., 1959. Testing Statistical Hypotheses. John Wiley & Sons, New York.Google Scholar
  8. Malley, J.D., 1980. Optimal unbiased estimation of variance components. Lecture Notes in Statistics v 39. Springer Verlag, New York.Google Scholar
  9. Rao, C., Rao, B.M., 1998. Matrix Algebra and Its Applications to Statistics and Econometrics. World Scientific, London.CrossRefGoogle Scholar
  10. Seely, J., 1970. Linear spaces and unbiased estimators-Application to the mixed linear model. The Annals of Mathematical Statistics,41(5), 1735–1745.MathSciNetCrossRefGoogle Scholar
  11. Seely, J., 1971. Quadratic subspaces and completeness. The Annals of Mathematical Statistics,42(2), 710–721.MathSciNetCrossRefGoogle Scholar
  12. Seely, J., Zyskind, G., 1971. Linear spaces and minimum variance estimators. The Annals of Mathematical Statistics,42(2), 691–703.CrossRefGoogle Scholar
  13. Vanleeuwen, D., Birkes, D., Seely, J., 1999. Balance and orthogonality in designs for mixed classification models. The Annals of Statistics,27(6), 1927–1947.MathSciNetCrossRefGoogle Scholar
  14. Vanleeuwen, D., Seely, J., Birkes, D., 1998. Sufficient conditions for orthogonal designs in mixed linear models. Journal of Statistical Planning and Inference, 73, 373–389.MathSciNetCrossRefGoogle Scholar
  15. Wulff, S.S., Birkes, D., 2005. Minimum variance unbiased invariant estimation of variance components under normality. Statistics, 39, 53–65.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2010

Authors and Affiliations

  • S. S. Ferreira
    • 1
    Email author
  • D. Ferreira
    • 1
  • C. Nunes
    • 1
  • J. T. Mexia
    • 2
  1. 1.Mathematics DepartmentUniversity of Beira InteriorCovilhãPortugal
  2. 2.Mathematics Department, Faculty of Science and TechnologyNew University of LisbonCaparicaPortugal

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