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GENERATING FUNCTIONS AND SHORT RECURSIONS, WITH APPLICATIONS TO THE MOMENTS OF QUADRATIC FORMS IN NONCENTRAL NORMAL VECTORS

Published online by Cambridge University Press:  17 October 2013

Grant Hillier ,
Raymond Kan  and
Xiaolu Wang
Grant Hillier
Affiliation:
CeMMAP University of Southampton
Raymond Kan*
Affiliation:
University of Toronto
Xiaolu Wang
Affiliation:
Iowa State University
*
*Address correspondence to Raymond Kan, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6, e-mail: [email protected].
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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, dk, say, is expressed in terms of all lower order dj’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.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 2 , April 2014 , pp. 436 - 473
Copyright
Copyright © Cambridge University Press 2013 

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