Computationally efficient recursions for top-order invariant polynomials with applications

Grant Hillier, Raymond Kan and Xiaolu Wang
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Raymond Kan: Institute for Fiscal Studies
Xiaolu Wang: Institute for Fiscal Studies

No CWP07/08, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies

Abstract:

The top-order zonal polynomials Ck(A),and top-order invariant polynomials Ck1,...,kr(A1,...,Ar)in which each of the partitions of ki,i = 1,..., r,has only one part, occur frequently in multivariate distribution theory, and econometrics - see, for example Phillips (1980, 1984, 1985, 1986), Hillier (1985, 2001), Hillier and Satchell (1986), and Smith (1989, 1993). However, even with the recursive algorithms of Ruben (1962) and Chikuse (1987), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.

JEL-codes: C16 C46 C63 (search for similar items in EconPapers)
Date: 2008-02-20
New Economics Papers: this item is included in nep-ecm
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Related works:
Journal Article: COMPUTATIONALLY EFFICIENT RECURSIONS FOR TOP-ORDER INVARIANT POLYNOMIALS WITH APPLICATIONS (2009)
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