Invariant Polynomials and Related Tests

Abstract

Many test statistics for testing hypotheses on the parameters of one or more multinormal populations are associated with the latent roots of certain positive definite symmetric matrices. Thus the distributions of latent roots play vital roles in some testing problems. One such case is the joint distribution of the latent roots of a Wishart matrix. One representation of this joint density is in terms of zonal polynomials. Zonal polynomials are certain homogeneous polynomials in the latent roots of a symmetric matrix, which are invariant under orthogonal transformations. Such invariant polynomials and related tests are examined in this article. Some applications of these topics such as multivariate calibration, regression models, discriminant analysis, robustness studies and econometric theory are also considered.