Tests of Hypotheses for Covariance Matrices and Distributions Under Multivariate Normal Populations

SYNOPTIC ABSTRACT

This paper treats the following tests of hypotheses on covariance matrices under multivariate normal distributions: (i) the equality of a covariance matrix to a given matrix, (ii) sphericity, (iii) independence, (iv) equality of covariance matrices, and (v) equality of covariance matrices against ‘one–sided’ alternatives. Although we have several test criteria for (v), the test criteria for the other problems are the likelihood ratio (LR) criteria the exact distributions of which are considered in the early 1970's. The main purpose of this paper is to review test criteria introduced by the author for testing hypotheses about a covariance matrix. We give the likelihood ratio criteria and forms of their distributions. These forms are asymptotic expansions that are different under the null hypothesis and the fixed alternatives, except for problem (v) for which these forms are the same.

Key Words and Phrases:

• hypotheses for covariance matrices
• equality of covariance matrices
• sphericity test
• test for independence
• one–sided alternatives
• likelihood ratio criteria
• asymptotic expansions