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On the construction of a class of invariant polynomials in several matrices, extending the zonal polynomials

  • A. W. Davis
Article

Summary

The construction of a class of invariant polynomials in several matrices extending the zonal polynomials is discussed. The method adopted generalized the orginal group-theoretic approach of James [9]. A table of three-matrix polynomials up to degree 5 is presented.

AMS 1970 subject classification

62E15 62H10 

Key words and phrases

Invariant polynomials zonal polynomials group representations multivariate distributions Young tableaux 

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References

  1. [1]
    Boerner, H. (1963).Representations of Groups, North-Holland, Amsterdam.zbMATHGoogle Scholar
  2. [2]
    Chikuse, Y. (1979a). Distributions of some matrix variates and latent roots in multivariate Behrens-Fisher discriminant analysis, to appear inAnn. Statist.Google Scholar
  3. [3]
    Chikuse, Y. (1979b). Invariant polynomials with three matrix arguments, extending the polynomials with smaller numbers of matrix arguments, unpublished report.Google Scholar
  4. [4]
    Chikuse, Y. (1980). Invariant polynomials with real and complex matrix arguments and their applications, unpublished report, University of Pittsburgh.Google Scholar
  5. [5]
    Constantine, A. G. (1963). Some non-central distribution problems in multivariate analysis,Ann. Math. Statist.,34, 1270–1285.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Davis, A. W. (1979). Invariant polynomials with two matrix arguments extending the zonal polynomials: applications to multivariate distribution theory.Ann. Inst. Statist. Math.,31, A, 465–485.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Davis, A. W. (1980a). Invariant polynomials with two matrix arguments, extending the zonal polynomials,Multivariate Analysis—V (ed. P. R. Krishnaiah), 287–299.Google Scholar
  8. [8]
    Davis, A. W. (1980b). On the effects of moderate multivariate nonnormality on Wilks's likelihood ratio criterion,Biometrika,67, 419–427.MathSciNetCrossRefGoogle Scholar
  9. [9]
    James, A. T. (1961a). Zonal polynomials of the real positive definite symmetric matrices,Ann. Math.,74, 456–469.MathSciNetCrossRefGoogle Scholar
  10. [10]
    James, A. T. (1961b). The distribution of noncentral means with known covariance,Ann. Math. Statist.,32, 874–882.MathSciNetCrossRefGoogle Scholar
  11. [11]
    James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples,Ann. Math. Statist.,35, 475–501.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Muirhead, R. J. (1978). Latent roots and matrix variates: a review of some asymptotic results,Ann. Statist.,6, 5–33.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Phillips, P. C. B. (1980). The exact distribution of instrumental variable estimators in an equation containingn+1 endogenous variables,Econometrica,48, 861–878.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Richards, D. St. P. and Gupta, R. D. (1980). Evaluation of cumulative probabilities for Wishart and multivariate beta matrices and their latent roots, unpublished report, University of the West Indies.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1981

Authors and Affiliations

  • A. W. Davis

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