Generating Functions and Short Recursions, with Applications to the Moments of Quadratic Forms in Noncentral Normal Vectors

Abstract

Recursive relations for objects of statistical interest have long been important for computation, and 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), Hillier, Kan, and Wang (2009)). Typically, in a recursion of this type the k-th object of interest, d_k say, is expressed in terms of all lower-order d_j's. In Hillier, Kan, and Wang (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, Kan, and Wang (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 (c.f. Brown (1986)), producing a super-short recursion that is significantly more efficient than even the short recursion.

Keywords: Generating functions, Quadratic forms, Ratio of quadratic forms

JEL Classification: E10, H10