Skip to main content
SpringerLink
Log in

Asymptotic formulas for the non-null distributions of three statistics for multivariate linear hypothesis

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Summary

Asymptotic formulas for the distributions of the likelihood ratio statistic, Hotelling's statistic and Pillai's statistic for multivariate linear hypothesis are derived under the assumption ofn e =ne, n h =nh (e>0, h>0, e+h=1) and Ω=O(n), wheren e andn h are the degrees of freedom for the error and for the hypothesis, respectively and Ω is the non-centrality matrix. New asymptotic formulas are given in terms of normal distribution function and its derivatives up to the ordern −1. We give also some numerical results of our asymptotic approximations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, T. W. (1946). The non-central Wishart distribution and certain problems of multivariate statistics,Ann. Math. Statist.,17, 409–431.

    Google Scholar 

  2. Fujikoshi, Y. (1968). Asymptotic expansion of the distribution of the generalized variance in the non-central case.J. Sci. Hiroshima Univ, Ser. A-I,32, 293–299.

    MathSciNet  Google Scholar 

  3. Fujikoshi, Y. (1970). Asymptotic expansions of the distributions of the test statistics in multivariate analysis,J. Sci. Hiroshima Univ., Ser. A-I,34, 73–144.

    MathSciNet  Google Scholar 

  4. Lee, Y. S. (1971). Distribution of the canonical correlations and asymptotic expansions for the distributions of certain independence test statistics,Ann. Math. Statist.,42, 526–537.

    MathSciNet  Google Scholar 

  5. Lee, Y. S. (1971). Asymptotic formulae for the distribution of a multivariate test statistic: power comparisons of certain multivariate tests,Biometrika,58, 647–651.

    Article  Google Scholar 

  6. Muirhead, R. J. (1972). The asymptotic noncentral distribution Hotelling's generalizedT 20 ,Ann. Math. Statist.,43, 1671–1677.

    MathSciNet  Google Scholar 

  7. Nagao, H. (1970). Asymptotic expansions of some test criteria for homogeneity of variances and covariance matrices from normal populations,J. Sci. Hiroshima Univ., Ser. A-I,34, 153–247.

    MathSciNet  Google Scholar 

  8. Pillai, K. C. S. and Jayachandran, K. (1967). Power comparisons of tests of two multivariate hypotheses based on four criteria,Biometrika,54, 195–210.

    Article  MathSciNet  Google Scholar 

  9. Siotani, M. (1971). An asymptotic expansion of the non-null distribution of Hotelling's generalizedT 20 -statistic,Ann. Math. Statist.,42, 560–571.

    MathSciNet  Google Scholar 

  10. Sugiura, N. (1969). Asymptotic expansions of the distributions of |S 1 S 2 −1| and |S 2(S 1+S 2)−1| for testing the equality of two covariance matrices,Mimeo Series, No. 653, Univ. of North Carolina.

  11. Sugiura, N. and Fujikoshi, Y. (1969). Asymptotic expansions of the non-null distributions of the likelihood ratio criteria for multivariate linear hypothesis and independence,Ann. Math. Statist.,40, 942–952.

    MathSciNet  Google Scholar 

  12. Sugiura, N. (1973). Further asymptotic formulas for the non-null distributions of three statistics for multivariate linear hypothesis,Ann. Inst. Statist. Math.,25, 153–163.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kobe University, Kobe, Japan

    Yasunori Fujikoshi

Authors
  1. Yasunori Fujikoshi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was partially supported by the Sakko-Kai Foundation.

About this article

Cite this article

Fujikoshi, Y. Asymptotic formulas for the non-null distributions of three statistics for multivariate linear hypothesis. Ann Inst Stat Math 27, 99–108 (1975). https://doi.org/10.1007/BF02504629

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02504629

Keywords

Search

Navigation