Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors
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 (1962), Hillier, Kan, and Wang (2009)). Typically, in a recursion of this type the k-th object of interest, dk say, is expressed in terms of all lower-order dj â€™s. In Hillier, Kan, and Wang (2009) 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 Hillier, Kan, and Wang (2009) 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 Keywords; generating functions, invariant polynomials, non-central normal distribution, recursions, symmetric functions, zonal polynomials
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- Chikuse, Yasuko, 1987. "Methods for Constructing Top Order Invariant Polynomials," Econometric Theory, Cambridge University Press, vol. 3(02), pages 195-207, April.
- Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Other publications TiSEM c6725407-ac3c-44fd-b6d1-5, Tilburg University, School of Economics and Management.
- 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.
- 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.
- 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.
- Ghazal, G. A., 1994. "Moments of the ratio of two dependent quadratic forms," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 313-319, July.
- Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
- 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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