Computationally Efficient Recursions for Top-Order Invariant Polynomials with Applications

39 Pages Posted: 20 Mar 2007 Last revised: 1 Mar 2008

See all articles by Raymond Kan

Raymond Kan

University of Toronto - Rotman School of Management

Xiaolu Wang

Iowa State University

Grant Hillier

University of Southampton - Division of Economics

Abstract

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.

Keywords: Invariant polynomials, quadratic form

JEL Classification: C10, C63

Suggested Citation

Kan, Raymond and Wang, Xiaolu and Hillier, Grant, Computationally Efficient Recursions for Top-Order Invariant Polynomials with Applications. Econometric Theory, Forthcoming, Available at SSRN: https://ssrn.com/abstract=968146 or http://dx.doi.org/10.2139/ssrn.968146

Raymond Kan (Contact Author)

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S3E6
Canada
416-978-4291 (Phone)
416-971-3048 (Fax)

Xiaolu Wang

Iowa State University ( email )

2167 Union Drive
Ames, IA 50011
United States

Grant Hillier

University of Southampton - Division of Economics ( email )

Southampton, SO17 1BJ
United Kingdom
+44 02380 592659 (Phone)