Methods for Constructing Top Order Invariant Polynomials

Abstract

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 ki 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.

Computationally efficient recursions for top-order invariant polynomials with applications

“…, r, with H an n × n orthogonal matrix. Chikuse (1987) provides an explicit expression for d κ in terms of the p κ as…”

“…, r, has only one part, occur frequently in multivariate distribution theory, and econometrics -see, for example Phillips (1980), Hillier (1985, 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.…”

“…For the case of the top-order invariant polynomials with several arguments, Chikuse (1987) presents both explicit and recursive expressions for the polynomials. Unlike the recursive relation for top-order zonal polynomials, the expressions in Chikuse (1987) are difficult to implement, and also very time consuming.…”

“…Unlike the recursive relation for top-order zonal polynomials, the expressions in Chikuse (1987) are difficult to implement, and also very time consuming. For the two matrix arguments case, Smith (1993) simplifies the explicit expression of top-order invariant polynomials when the order in one of the terms is just 1 or 2.…”

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

“…These expressions follow easily from the moment generating function (MGF) of q, and the joint moment generating function of the q i , both of which have expansions in terms of these polynomials (see below). Ruben [19] and James [10] essentially give (1), while Chikuse [3] gives (2). The density function of q may also be expressed as an infinite series in the C k (A), see James [10], Ruben [19], and Section 4 below.…”