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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, 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.
CWP14/08
Centre for Microdata Methods and Practice
Hillier, Grant
3423bd61-c35f-497e-87a3-6a5fca73a2a1
Kan, Raymond
4068dcb5-18f4-4e95-845c-88e5e458fcfa
Wang, Xiaolu
14400710-0506-40ae-b362-ad38333cdf9c
Hillier, Grant
3423bd61-c35f-497e-87a3-6a5fca73a2a1
Kan, Raymond
4068dcb5-18f4-4e95-845c-88e5e458fcfa
Wang, Xiaolu
14400710-0506-40ae-b362-ad38333cdf9c

Hillier, Grant, Kan, Raymond and Wang, Xiaolu (2008) Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors (CEMMAP Working Paper, CWP14/08) London, GB. Centre for Microdata Methods and Practice

Record type: Monograph (Working 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, 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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Published date: June 2008

Identifiers

Local EPrints ID: 55071
URI: http://eprints.soton.ac.uk/id/eprint/55071
PURE UUID: f63d3743-d8b3-44a0-b1b3-739932b5f5c4
ORCID for Grant Hillier: ORCID iD orcid.org/0000-0003-3261-5766

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Date deposited: 06 Aug 2008
Last modified: 14 Dec 2023 02:33

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Contributors

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

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