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

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

generating functions, invariant polynomials, non-central normal distribution, recursions, symmetric functions, zonal polynomials
0966-4246
918
University of Southampton
Hillier, Grant
3423bd61-c35f-497e-87a3-6a5fca73a2a1
Kan, Raymond
4068dcb5-18f4-4e95-845c-88e5e458fcfa
Wang, Xiaoulu
9eaa9931-c016-41aa-b243-4a34fdfe2d5f
Hillier, Grant
3423bd61-c35f-497e-87a3-6a5fca73a2a1
Kan, Raymond
4068dcb5-18f4-4e95-845c-88e5e458fcfa
Wang, Xiaoulu
9eaa9931-c016-41aa-b243-4a34fdfe2d5f

Hillier, Grant, Kan, Raymond and Wang, Xiaoulu (2009) Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors (Discussion Papers in Economics and Econometrics, 918) Southampton. University of Southampton 45pp.

Record type: Monograph (Discussion Paper)

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 (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 ’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

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More information

Published date: 2009
Keywords: generating functions, invariant polynomials, non-central normal distribution, recursions, symmetric functions, zonal polynomials

Identifiers

Local EPrints ID: 71072
URI: http://eprints.soton.ac.uk/id/eprint/71072
ISSN: 0966-4246
PURE UUID: cca83d6f-45de-4bfd-bb78-07b95e8ee6a0
ORCID for Grant Hillier: ORCID iD orcid.org/0000-0003-3261-5766

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Date deposited: 15 Jan 2010
Last modified: 14 Mar 2024 02:36

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Contributors

Author: Grant Hillier ORCID iD
Author: Raymond Kan
Author: Xiaoulu Wang

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