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

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  • Grant Hillier

    ()
    (Institute for Fiscal Studies and University of Southampton)

  • Raymond Kan
  • Xiaolu Wang

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.

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Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP14/08.

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Date of creation: Jun 2008
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Handle: RePEc:ifs:cemmap:14/08

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  1. Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153219, Tilburg University.
  2. Grant Hillier & Raymond Kan & Xiaolu Wang, 2008. "Computationally efficient recursions for top-order invariant polynomials with applications," CeMMAP working papers CWP07/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  3. Sawa, Takamitsu, 1972. "Finite-Sample Properties of the k-Class Estimators," Econometrica, Econometric Society, vol. 40(4), pages 653-80, July.
  4. Chikuse, Yasuko, 1987. "Methods for Constructing Top Order Invariant Polynomials," Econometric Theory, Cambridge University Press, vol. 3(02), pages 195-207, April.
  5. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
  6. 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.
  7. Ghazal, G. A., 1994. "Moments of the ratio of two dependent quadratic forms," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 313-319, July.
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