Vol.13 No.1 ACTA MATHEMATICAE APPLICATAE SINICA Jan., 1997 THE EXPECTED VALUES OF INVARIANT POLYNOMIALS WITH MATRIX ARGUMENT OF ELLIPTICAL DISTRIBUTIONS* LI RUNZE (~j~) (Probability Laboratory, Institute of Applied Mathematics, the Chinese Academy of Sciences, Beijing 100080, China) Abstract Invariant polynomials with matrix arguments have been defined by the theory of group representation, generalizing the zonal polynomials. -They have developed as a useful tool to evaluate certain integrals arising in multivariate distribution theory, which were expanded as power series in terms of the invariant polynomials. Some interesting polynomials has been shown by people working in the field of econometric theory. In this paper, we derive the expected values of C~' ~(/~gB~, C~ (BR)Ca (~) and C~ (B- 1U), where Bd----X'X and X~ x p is distributed according to an elliptical matrix distribution. We also give their applications in multivariate distribution theory including the related development in econometrics. Key words. Elliptical matrix distributions, invariant polynomials, zonal polynomials 1. Introduction There are many publications on matrix variate distributions (or matrix distribution for simplicity), in particular, about their expected values of zonal polynomials of their quadratic forms (cf. [12], [15], [17] and [18]). However, there remain distributional problems which cannot be solved in terms of the zonal polynomials, including noncentral distributions with more matrix parameters, some of which arise, in particular, from a formal approach to nonnormal distributions in multivariate Edgeworth populations (cf. [6]). An extension of the zonal polynomials to invariant polynomials C~Er](X[~]) in r matrices, where X[~| = (X1,--- ,X~) has been given for r = 2 by Davis[ vt and for r > 3 by Chikuse [2]. For r = 2, we write C~'A(X,~ instead of C~[2](X,Y). The property of C~'A(X,Y) has been stated in [7]. The usefulness of the invariant polynomials has been recognized in deriving series expansions of the exact distributions of estimators and test statistics in simultaneous equations models, and substantials progress has been made in econometrics theory in recent years (cf. [3] and [4]). Received July 30, 1993. Revised February 21, 1995. * This research is supported by the Chinese Academy of Sciences.