The VDR Test About The Covariance Matrix Of Normal Populations

Hypothesis testing for the covariance matrix is an important aspect of statistical inference in the multivariate analysis, which has been widely used in many fields. There were many tests about the covariance matrix include the classical likelihood-ratio type tests, tests based on the empirical distance, tests based on the largest eigenvalue of the covariance matrix and tests based on random matrix theory. But they are asymptotic tests whose test statistics' exact distributions under null hypothesis are difficult to calculate. In this paper, an exact test about the covariance matrix of Normal populations based on VDR(Vertical Density Representation) test was proposed.The VDR test is based on randomized estimation of parameter and VDR theory, which is a universal method and could be applied to many fields. The confidence region corresponds to VDR test is the best with respect to Lebesgue measure under some conditions. Applying VDR test method to test the covariance matrix of Normal populations had deep theoretical foundation. In this paper, two types of hypotheses about the covariance matrix of Normal populations were tested, one is for the equality of covariance matrix to a given matrix, and the other is for covariance structure. We also analyzed whether the VDR test works for the first type when the dimensionality p is large, and in particular not less than the sample size n. Via an extensive simulation study the VDR test was compared with the classical likelihood-ratio type tests and other tests about the covariance matrix. The VDR test performed perfectly on the control of the type?error rate, which is the advantage of the exact test. We showed that the VDR test is relatively powerful and robust in each case via this simulation study. The VDR test is an effective method to test hypotheses about the covariance matrix of Normal populations.