Methods for Constructing Top Order Invariant Polynomials
给出了三种构造最高阶不变多项式的方法，包括两种显式公式和一种递推程序，这些多项式在多元分布理论中频繁出现，对计量经济学中的分布理论应用有帮助。
The invariant polynomials (Davis [8] and Chikuse [2] with r ( r ≥ 2) symmetric matrix arguments have been defined, extending the zonal polynomials, and applied in multivariate distribution theory. The usefulness of the polynomials has attracted the attention of econometricians, and some recent papers have applied the methods to distribution theory in econometrics (e.g., Hillier [14] and Phillips [22]). The ‘top order’ invariant polynomials , in which each of the partitions of k i 1 = 1,…, r , and has only one part, occur frequently in multivariate distribution theory (e.g., Hillier and Satchell [17] and Phillips [27]). In this paper we give three methods of constructing these polynomials, extending those of Ruben [28] for the top order zonal polynomials. The first two methods yield explicit formulae for the polynomials and then we give a recurrence procedure. It is shown that some of the expansions presented in Chikuse and Davis [4] are simplified for the top order invariant polynomials. A brief discussion is given on the ‘lowest order’ invariant polynomials.