Computationally Efficient Recursions for Top-Order Invariant Polynomials with Applications

The top-order zonal polynomials $C_{k}(A)$, and top-order invariant polynomials $C_{k_1,\ldots, k_r} (A_1,\ldots, A_r)$ in which each of the partitions of $k_i$, $i=1, \ldots, r$, 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