Skip to main content
SpringerLink
Log in

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

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

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.

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. Boerner, H. (1963).Representations of Groups, North-Holland, Amsterdam.

    MATH  Google Scholar 

  2. Chikuse, Y. (1979a). Distributions of some matrix variates and latent roots in multivariate Behrens-Fisher discriminant analysis, to appear inAnn. Statist.

  3. Chikuse, Y. (1979b). Invariant polynomials with three matrix arguments, extending the polynomials with smaller numbers of matrix arguments, unpublished report.

  4. Chikuse, Y. (1980). Invariant polynomials with real and complex matrix arguments and their applications, unpublished report, University of Pittsburgh.

  5. Constantine, A. G. (1963). Some non-central distribution problems in multivariate analysis,Ann. Math. Statist.,34, 1270–1285.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  7. Davis, A. W. (1980a). Invariant polynomials with two matrix arguments, extending the zonal polynomials,Multivariate Analysis—V (ed. P. R. Krishnaiah), 287–299.

  8. Davis, A. W. (1980b). On the effects of moderate multivariate nonnormality on Wilks's likelihood ratio criterion,Biometrika,67, 419–427.

    Article  MathSciNet  Google Scholar 

  9. James, A. T. (1961a). Zonal polynomials of the real positive definite symmetric matrices,Ann. Math.,74, 456–469.

    Article  MathSciNet  Google Scholar 

  10. James, A. T. (1961b). The distribution of noncentral means with known covariance,Ann. Math. Statist.,32, 874–882.

    Article  MathSciNet  Google Scholar 

  11. James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples,Ann. Math. Statist.,35, 475–501.

    Article  MathSciNet  Google Scholar 

  12. Muirhead, R. J. (1978). Latent roots and matrix variates: a review of some asymptotic results,Ann. Statist.,6, 5–33.

    Article  MathSciNet  Google Scholar 

  13. Phillips, P. C. B. (1980). The exact distribution of instrumental variable estimators in an equation containingn+1 endogenous variables,Econometrica,48, 861–878.

    Article  MathSciNet  Google Scholar 

  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.

Download references

Authors
  1. A. W. Davis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

CSIRO

About this article

Cite this article

Davis, A.W. On the construction of a class of invariant polynomials in several matrices, extending the zonal polynomials. Ann Inst Stat Math 33, 297–313 (1981). https://doi.org/10.1007/BF02480943

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02480943

AMS 1970 subject classification

Key words and phrases

Search

Navigation