Behavior of Some Hypothesis Tests for the Covariance Matrix of High Dimensional Data

The study of the structure of the covariance matrix when the dimension of the data is much greater than the sample size (high dimensional data) is a complicated problem, since we have many unknown parameters and few data. Several hypothesis tests for the covariance matrix, in the high dimensional context and in the classical case (where the dimension of the data is less than the sample size), can be found in the literature. It has been of interest the tests for the null hypothesis that the covariance matrix of Gaussian data is equal or proportional to the identity matrix, considering the classical case as well as the high dimensional context. Since it is important to have a wide comparison between these tests found in the literature, and for some of them it is difficult to have theoretical results about their powers, in this work we compare several tests by simulations, in terms of the size and power of the test. We also present some examples of application with real high dimensional data found in the literature.