Generalized F Tests in Models with Random Perturbations: The Truncated Normal Case

  • Célia Nunes
  • Dário Ferreira
  • Sandra Ferreira
  • João T. Mexia
Conference paper
Part of the Studies in Theoretical and Applied Statistics book series (STAS)


This paper shows how to obtain explicit expressions for non-central generalized F distributions with random non-centrality parameters. We consider the case when these parameters are random variables with truncated Normal distribution, for the usual F distribution and for the generalized F distribution.


  1. 1.
    Davies, R.B.: Algorithm AS 155: The distribution of a linear combinations of χ 2 random variables. Appl. Stat. 29, 232–333 (1980)Google Scholar
  2. 2.
    Fonseca, M., Mexia, J.T., Zmyślony, R.: Exact distribution for the generalized F tests. Discuss. Math. Probab. Stat. 22, 37–51 (2002)MathSciNetMATHGoogle Scholar
  3. 3.
    Gaylor, D.W., Hopper, F.N.: Estimating the degrees of freedom for linear combinations of mean squares by Satterthwaite’s formula. Technometrics 11, 691–706 (1969)CrossRefGoogle Scholar
  4. 4.
    Imhof, J.P.: Computing the distribution of quadratic forms in normal variables. Biometrika 48, 419–426 (1961)MathSciNetMATHGoogle Scholar
  5. 5.
    Michalski, A., Zmyślony, R.: Testing hypothesis for variance components in mixed linear models. Statistics 27, 297–310 (1996)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Michalski, A., Zmyślony, R.: Testing hypothesis for linear functions of parameters in mixed linear models. Tatra Mt. Math. Publs. 17, 103–110 (1999)MATHGoogle Scholar
  7. 7.
    Nunes, C., Mexia, J.T.: Non-central generalized F distributions. Discuss. Math. Probab. Stat. 26, 47–61 (2006)MathSciNetMATHGoogle Scholar
  8. 8.
    Nunes, C., Ferreira, S., Ferreira, D.: Generalized F tests in models with random perturbations: The Gamma case. Discuss. Math. Probab. Stat. 29(2), 185–198 (2009)MathSciNetMATHGoogle Scholar
  9. 9.
    Nunes, C., Ferreira, D., Ferreira, S., Mexia, J.T.: Generalized F distributions with random non-centrality parameters: The convolution of Gamma and Beta variables. Far East J. Math. Sci. 62(1), 1–14 (2012)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Robbins, H.: Mixture of distribution. Ann. Math. Statist. 19, 360–369 (1948)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Robbins, H., Pitman, E.J.G.: Application of the method of mixtures to quadratic forms in normal variates. Ann. Math. Statist. 20, 552–560 (1949)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Satterthwaite, F.E.: An approximate distribution of estimates of variance components. Biometrics Bull. 2, 110–114 (1946)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Célia Nunes
    • 1
  • Dário Ferreira
    • 1
  • Sandra Ferreira
    • 1
  • João T. Mexia
    • 2
  1. 1.Department of MathematicsUniversity of Beira InteriorCovilhãPortugal
  2. 2.Department of MathematicsNew University of LisbonCaparicaPortugal

Personalised recommendations