A Discussion for Test of Covariance Matrix with Likelihood Ratio Principle

In this study, we consider to discuss and propose a test procedure for covariance matrix under the normality assumption. For this purpose, first of all, we identify that the likelihood ratio function consists of a product of likelihood ratio functions of individual eigenvalues of covariance matrix. Then we construct a union-intersection type of statistic based on the minimum among individual p-values and propose a test. We note that the null distributions of the individual statistics are all chi-square and independent. Since the union-intersection test statistic is one of combination functions, we propose two more tests using the combination functions by combining p-values of individual partial tests. Then we compare the efficiency of the proposed tests with the asymptotic one by obtaining empirical powers through a simulation study. For this simulation study, we include the likelihood ratio test with chi-square which is the limiting distribution of the likelihood ratio statistic. Finally, we discuss some interesting features related with the test for the covariance matrix.