The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

  • K. C. S. Pillai
  • Yu-Sheng Hsu
Article

Summary

The joint density function of the latent roots ofS 1 S 2 −1 under violations is obtained whereS 1 has a complex non-central Wishart distributionW c (p,n 1,Σ 1,Ω) andS 2, an independent complex central Wishart,W c (p,n 2,Σ 2, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.

AMS classifications

62H10 62H15 

Key words and phrases

Complex Gaussian Characteristic roots distribution under violations non-normality unequal covariance matrices Hotelling's trace Pillai's trace Wilks' criterion Roy's largest root power comparisons tests of three hypotheses tabulations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Anderson, T. W. and Das Gupta, S. (1964). Monotonicity of the power functions of some tests of independence between two sets of variates,Ann. Math. Statist.,35, 206–208.MATHMathSciNetGoogle Scholar
  2. [2]
    Anderson, T. W. and Das Gupta, S. (1964). A monotonicity property of the power functions of some tests of the equality of two covariance matrices,Ann. Math. Statist.,35, 1059–1063.MathSciNetGoogle Scholar
  3. [3]
    Das Gupta, S., Anderson, T. W. and Mudholkar, G. S. (1964). Monotonicity of the power functions of some tests of the multivariate linear hypothesis,Ann. Math. Statist.,35, 200–205.MATHMathSciNetGoogle Scholar
  4. [4]
    Eaton, M. L. and Perlman, M. D. (1974). A monotonicity property of the power functions of some invariant tests for MANOVA,Ann. Statist.,2, 1022–1028.MATHMathSciNetGoogle Scholar
  5. [5]
    Ghosh, M. N. (1964). On the admissibility of some tests of MANOVA,Ann. Math. Statist.,35, 789–794.MATHMathSciNetGoogle Scholar
  6. [6]
    Herz, C. S. (1955). Bessel functions of matrix argument,Ann. Math.,61, 474–523.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    James, A. T. (1964). Distribution of matrix variates and latent roots derived from normal samples,Ann. Math. Statist.,35, 475–501.MATHMathSciNetGoogle Scholar
  8. [8]
    Khatri, C. G. (1965). Classical statistical analysis based on a certain multivariate complex Gaussian distribution,Ann. Math. Statist.,36, 98–114.MATHMathSciNetGoogle Scholar
  9. [9]
    Khatri, C. G. (1966). On certain distribution problems based on positive definite quadratic functions in normal vectors,Ann. Math. Statist.,37, 468–479.MATHMathSciNetGoogle Scholar
  10. [10]
    Khatri, C. G. (1969). Non-central distributions of theith largest characteristic roots of three matrices concerning complex multivariate normal populations,Ann. Inst. Statist. Math.,21, 23–32.MATHMathSciNetGoogle Scholar
  11. [11]
    Khatri, C. G. (1970). On the moments of traces of two matrices in three situations for complex multivariate normal populations,Sankhyã,32, 65–80.MATHMathSciNetGoogle Scholar
  12. [12]
    Kiefer, J. and Schwartz, R. (1965). Admissible character ofT 2,R 2, and other fully invariant tests for classical multivariate normal problems,Ann. Math. Statist.,36, 747–770.MATHMathSciNetGoogle Scholar
  13. [13]
    Mikhail, W. F. (1962). On a property of a test for the equality of two normal dispersion matrices against one-sided alternatives,Ann. Math. Statist.,33 1463–1465.MATHMathSciNetGoogle Scholar
  14. [14]
    Mudholkar, G. S. (1965). A class of tests with monotone power functions for two problems in multivariate statistical analysis,Ann. Math. Statist.,36, 1794–1801.MATHMathSciNetGoogle Scholar
  15. [15]
    Perlman, M. D. (1974). On the monotonicity of the power functions of tests based on traces of multivariate beta matrices,J. Multivariate Anal.,4, 22–30.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Pillai, K. C. S. and Jayachandran, K. (1967). Power comparisons of tests of two multivariate hypothesis based on four criteria,Biometrika,54, 195–210.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Pillai, K. C. S. and Jayachandran, K. (1968). Power comparisons of tests of equality of two covariance matrices based on four criteria,Biometrika,55, 335–342.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Pillai, K. C. S. and Li, H. C. (1970). Monotonicity of the power functions of some tests of hypothesis concerning multivariate complex normal distributions,Ann. Inst. Statist. Math.,22, 307–318.MATHMathSciNetGoogle Scholar
  19. [19]
    Pillai, K. C. S. (1975). The distribution of the characteristic roots ofS 1 S 2−1 under violations,Ann. Statist.,3, 773–779.MATHMathSciNetGoogle Scholar
  20. [20]
    Pillai, K. C. S. and Sudjana (1975). Exact robustness studies of tests of two multivariate hypothesis based on four criteria and their distribution problems under violations,Ann. Statist.,3, 617–636.MATHMathSciNetGoogle Scholar
  21. [21]
    Pillai, K. C. S. and Hsu, Yu-Sheng (1975). The distribution of the characteristic roots ofS 1 S 2−1 under violations in the complex case and power comparisons of four tests,Mimeograph Series No. 430, Department of Statistics, Purdue University.Google Scholar
  22. [22]
    Pillai, K. C. S. and Hsu, Yu-Sheng (1975). Exact robustness studies of the test of independence based on four multivariate criteria and their distribution problems under violations,Ann. Inst. Statist. Math.,31, A, 85–101.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Roy, S. N. and Mikhail, W. F. (1961). On the monotonic character of the power functions of two multivariate tests,Ann. Math. Statist.,32, 1145–1151.MATHMathSciNetGoogle Scholar
  24. [24]
    Schwartz, R. E. (1964). Properties of a test in MANOVA,Ann. Math. Statist.,35, 939–940.Google Scholar

Copyright information

© Kluwer Academic Publishers 1979

Authors and Affiliations

  • K. C. S. Pillai
    • 1
  • Yu-Sheng Hsu
    • 1
  1. 1.Purdue UniversityWest LafagetteIndia

Personalised recommendations