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Methods for Constructing Top Order Invariant Polynomials

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  • Chikuse, Yasuko

Abstract

The invariant polynomials (Davis [8] and Chikuse [2] with r(r ≥ 2) symmetric matrix arguments have been defined, extending the zonal polynomials, and applied in multivariate distribution theory. The usefulness of the polynomials has attracted the attention of econometricians, and some recent papers have applied the methods to distribution theory in econometrics (e.g., Hillier [14] and Phillips [22]).The ‘top order’ invariant polynomials , in which each of the partitions of ki 1 = 1,…,r, and has only one part, occur frequently in multivariate distribution theory (e.g., Hillier and Satchell [17] and Phillips [27]). In this paper we give three methods of constructing these polynomials, extending those of Ruben [28] for the top order zonal polynomials. The first two methods yield explicit formulae for the polynomials and then we give a recurrence procedure. It is shown that some of the expansions presented in Chikuse and Davis [4] are simplified for the top order invariant polynomials. A brief discussion is given on the ‘lowest order’ invariant polynomials.

Suggested Citation

  • Chikuse, Yasuko, 1987. "Methods for Constructing Top Order Invariant Polynomials," Econometric Theory, Cambridge University Press, vol. 3(2), pages 195-207, April.
    • Handle: RePEc:cup:etheor:v:3:y:1987:i:02:p:195-207_01
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    Cited by:

    1. Forchini, G., 2000. "The density of the sufficient statistics for a Gaussian AR(1) model in terms of generalized functions," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 237-243, November.
    2. Bao, Yong & Kan, Raymond, 2013. "On the moments of ratios of quadratic forms in normal random variables," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 229-245.
    3. Grant Hillier & Raymond Kan & Xiaolu Wang, 2008. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," CeMMAP working papers CWP14/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Hillier, Grant & Kan, Raymond & Wang, Xiaoulu, 2009. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," Discussion Paper Series In Economics And Econometrics 918, Economics Division, School of Social Sciences, University of Southampton.
    5. Hillier, Grant & Kan, Raymond & Wang, Xiaoulu, 2009. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," Discussion Paper Series In Economics And Econometrics 0918, Economics Division, School of Social Sciences, University of Southampton.
    6. Giovanni Forchini, "undated". "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.
    7. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(1), pages 211-242, February.

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