The distributions of the likelihood ratio statistics for tests of certain covariance structures of complex multivariate normal populations

Abstract

This paper examines approximations to the distributions of the likelihood ratio statistics for testing hypotheses about certain structures for the covariance matrices of complex multivariate normal populations. Some applications of these distributions to inference on multiple time series are also discussed. The covariance matrices of complex multivariate normal populations play an important role in studying the spectral density matrices of stationary Gaussian multiple time series, because certain estimates of these matrices have approximately complex Wishart distribu- tions. In this paper we study approximations to the distributions of the likelihood ratio tests for multiple independence, sphericity and multiple homogeneity of the covariance matrices, as well as the hypothesis that the covariance matrix is equal to a specified matrix. Using the first four moments, distributions of certain powers of the test statistics are approxi- mated by Pearson's type I distributions. The accuracy of the approximations is found to be sufficient for practical purposes.