ON ASYMPTOTIC DISTRIBUTIONS OF TEST STATISTICS FOR COVARIANCE MATRICES AND CORRELATION MATRICES

Abstract

This chapter presents the asymptotic distributions of test statistics for covariance matrices and correlation matrices. It describes approximations to the distributions of certain functions of the elements of the sample covariance matrix when the underlying distribution is a mixture of two multivariate normal populations. The matrix S is distributed as noncentral Wishart matrix with n degrees of freedom and the noncentrality matrix M. Approximations to the distributions of test statistics for several hypotheses on the covariance matrix can be obtained when the sample is contaminated with outliers and the distribution underlying the data is a mixture of multivariate normal distributions. The chapter also presents the use of the saddle-point approach to obtain the asymptotic distribution for a function of eigenvalues of a multivariate quadratic form. Tests on eigenvalues of the covariance matrix arise in principal component analysis and other areas.