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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

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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, dk say, is expressed in terms of all lower-order dj'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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Find related papers by JEL classification:
C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: 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 and Programming - - - Computational Techniques

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  1. Ghazal, G. A., 1994. "Moments of the ratio of two dependent quadratic forms," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 313-319, July. [Downloadable!] (restricted)
  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(01), pages 211-242, February. [Downloadable!]
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  3. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March. [Downloadable!] (restricted)
  4. 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. [Downloadable!] (restricted)
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