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Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

Author

Listed:
  • Grant Hillier

    (Institute for Fiscal Studies and University of Southampton)

  • Raymond Kan

    (Institute for Fiscal Studies)

  • Xiaolu Wang

    (Institute for Fiscal Studies)

Abstract

Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben, Hillier, Kan, and Wang). Typically, in a recursion of this type the k -th object of interest, d k say, is expressed in terms of all lower-order d j 's. In Hillier, Kan, and Wang we pointed out that, in the case of top-order zonal polynomials (and generalizations of them), a shorter (i.e., fixed length) recursion can be deduced. The present paper shows that the argument in generalizes to a large class of objects/generating functions. The results thus obtained are then applied to various problems involving quadratic forms in noncentral normal vectors.

Suggested Citation

  • 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.
    • Handle: RePEc:ifs:cemmap:14/08
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    File URL: http://cemmap.ifs.org.uk/wps/cwp1408.pdf
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    References listed on IDEAS

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    1. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    2. 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.
    3. R. W. Farebrother, 1984. "The Distribution of a Positive Linear Combination of X2 Random Variables," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(3), pages 332-339, November.
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    6. Chikuse, Yasuko, 1987. "Methods for Constructing Top Order Invariant Polynomials," Econometric Theory, Cambridge University Press, vol. 3(2), pages 195-207, April.
    7. Bock, M.E. & Judge, G.G. & Yancey, T.A., 1984. "A simple form for the inverse moments of non-central [chi]2 andF random variables and certain confluent hypergeometric functions," Journal of Econometrics, Elsevier, vol. 25(1-2), pages 217-234.
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    9. Sawa, Takamitsu, 1972. "Finite-Sample Properties of the k-Class Estimators," Econometrica, Econometric Society, vol. 40(4), pages 653-680, July.
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    JEL classification:

    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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