Likelihood Ratio Tests for Equality of Mean Vectors with Circular Covariance Matrices

Abstract

While the likelihood ratio test for the equality of mean vectors, when the covariance matrices are assumed to be only positive-definite, is a common test in multivariate analysis, similar likelihood ratio tests are not available in the literature when the covariance matrices are assumed to have some common given structure. In this compact paper the author deals with the problem of developing likelihood ratio tests for the equality of mean vectors when the covariance matrices are assumed to have a circular or circulant structure. The likelihood ratio statistic is obtained and its exact distribution is expressed in terms of products of independent Beta random variables. Then, it is shown how for some particular cases it is possible to obtain very manageable finite form expressions for the probability density and cumulative distribution functions of this distribution, while for the other cases, given the intractability of the expressions for these functions, very sharp near-exact distributions are developed. Numerical studies show the extreme closeness of these near-exact distributions to the exact distributions.