Computationally efficient recursions for top-order invariant polynomials with applications

Hillier, Grant, Kan, R. and Wang, Xiaolu (2009) Computationally efficient recursions for top-order invariant polynomials with applications. Econometric Theory, 25 (1), 211-242. (doi:10.1017/S0266466608090075).

Record type: Article

Abstract

The top-order zonal polynomials Ck(A), and top-order invariant polynomials Ck1, , Ar) in which each of the partitions of ki, i = 1, see, for example, Phillips (1980, Econometrica 48, 861 398; 1985, International Economic Review 26, 21 896), Hillier (1985, Econometric Theory 1, 53 28), Hillier and Satchell (1986, Econometric Theory 2, 66 257; 1993, Australian Journal of Statistics 35, 271 570) and Chikuse (1987, Econometric Theory 3, 195 207), 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.

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