The simultaneous test of equality and circularity of several covariance matrices

The equality of covariance matrices test is widely used in many statistical procedures, and recently, matrices with a circular pattern, which are particular cases of the well-known Toeplitz matrices, have been considered in practical applications. In this work we present a statistical procedure that may allow one to test simultaneously both the equality and the circularity of a given number of covariance matrices. This procedure is based on a likelihood ratio test that is derived through an adequate decomposition of the overall null hypothesis in two partial null hypotheses, and on near-exact approximations developed for the test statistic. Numerical studies are carried out to assess the quality of these approximations and to illustrate their asymptotic properties. Simulation studies are also developed to study the properties of test.