Global one-sample tests for high-dimensional covariance matrices

Testing high-dimensional covariance matrix plays an important role in multivariate statistical analysis. Many statisticians used the statistics based on tr[(Sigma Sigma(-1)(0)-I-p)(2)] to test H-0 : Sigma = Sigma(0) with Sigma being the covariance matrix. However, none have proposed a statistic based on tr[(Sigma - Sigma(0))(2)] for this purpose. In fact, neither of the two tests is superior to the other based on their powers because they target different kinds of dense alternatives. Furthermore, some maximum-type tests were proposed to accommodate sparse alternatives. By using the advantages of these tests, we propose two new tests for one-sample covariance matrix when the sample size and data dimension increase proportionally. One is suitable for dense alternatives, the other is powerful against a wide range of situations, such as dense, sparse or a mixture of both alternatives. Extensive simulation results show that our proposed tests maintain high powers against various alternatives while the existing tests fail in at least one situation.