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Some comments on zonal polynomials and their expected values with respect to elliptical distributions

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Abstract

In this paper, we give alternative proofs of some results in [15] (Li R.,1997) about the expected value of zonal polynomials.

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References

  1. Caro-Lopera, F.J., Díaz-García, J.A., González-Farías, G. Noncentral elliptical configuration density. J. Multivariate Anal., 101(1): 32–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Constantine, A.C. Noncentral distribution problems in multivariate analysis. Ann. Math. Statist., 34: 1270–1285 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  3. Constantine, A.C. The distribution of Hotelling’s generalised T 0 2. Ann. Math. Statist., 37: 215–225 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  4. Díaz-García, J.A., Gutiérrez-Jáimez, R. Zonal polynomials: A note. Chilean J. Statist., 2(1): 21–30 (2010)

    MATH  Google Scholar 

  5. Fang, K.T., Zhang, Y.T. Generalized Multivariate Analysis. Science Press, Springer-Verlag, Beijing, 1990

    MATH  Google Scholar 

  6. Farrell, R.H. Multivariate Calculation: Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York, 1985

    Book  MATH  Google Scholar 

  7. James, A.T. The distribution of the latent roots of the covariance matrix. Ann. Math. Statist., 31: 151–158 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  8. James, A.T. Zonal polynomials of the real positive definite symmetric matrices. Ann. Math., 74(2): 456–469 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Khatri, C.G. On certain distribution problems based on positive definite quadratic functions in normal vector. Ann. Math. Statist., 37: 468–479 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  12. Koev, P., Edelman, A. The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp., 75: 833–846 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mathai, A.M. Jacobians of matrix transformations and functions of matrix argument. World Scientific, London, 1997

    Book  MATH  Google Scholar 

  14. Muirhead, R.J. Aspects of Multivariate Statistical Theory. John Wiley & Sons, New York, 1982

    Book  MATH  Google Scholar 

  15. Li, R. The expected values of invariant polynomials with matrix argument of elliptical distributions. Acta Math. App. Sinica, 13: 64–70 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Takemura, A. Zonal Polynomials. IMS Lecture Notes-Monograph Series, 1984

    MATH  Google Scholar 

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Acknowledgements

The authors wish to express their gratitude to the editor and the referees for their helpful comments and suggestions. This paper was written during J.A. Díaz-García’s stay as a visiting professor at the Department of Statistics and O. R. of the University of Granada, Spain.

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Correspondence to José A. Díaz-García.

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This research work was partially supported by National Council of Science and Technology (CONACYT)-Mexico, research grant 81512 and by Research, Development and Innovation (IDI)-Spain, grant MTM2005-09209.

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Díaz-García, J.A., Gutiérrez-Jáimez, R. Some comments on zonal polynomials and their expected values with respect to elliptical distributions. Acta Math. Appl. Sin. Engl. Ser. 33, 567–574 (2017). https://doi.org/10.1007/s10255-011-0092-8

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  • DOI: https://doi.org/10.1007/s10255-011-0092-8

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