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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
Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962, Annals of Mathematical Statistics 33, 542–570; Hillier, Kan, and Wang, 2009, Econometric Theory 25, 211–242). Typically, in a recursion of this type the kth object of interest, d k , say, is expressed in terms of all lower order d j ’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, a fixed length recursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986, SIAM Journal on Scientific and Statistical Computing 7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.
436-473
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 (2014) Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors. Econometric Theory, 30 (2), 436-473. (doi:10.1017/S0266466613000364).

Record type: Article

Abstract

Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962, Annals of Mathematical Statistics 33, 542–570; Hillier, Kan, and Wang, 2009, Econometric Theory 25, 211–242). Typically, in a recursion of this type the kth object of interest, d k , say, is expressed in terms of all lower order d j ’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, a fixed length recursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986, SIAM Journal on Scientific and Statistical Computing 7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.

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e-pub ahead of print date: 17 October 2013
Published date: April 2014
Organisations: Economics

Identifiers

Local EPrints ID: 373103
URI: http://eprints.soton.ac.uk/id/eprint/373103
PURE UUID: 53a7e51f-3c42-4783-b7da-8a6be5a8e3eb
ORCID for Grant Hillier: ORCID iD orcid.org/0000-0003-3261-5766

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Date deposited: 07 Jan 2015 11:58
Last modified: 15 Mar 2024 02:44

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Contributors

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

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