Tests for covariance matrices in high dimension with less sample size

In this article, we propose tests for covariance matrices of high dimension with fewer observations than the dimension for a general class of distributions with positive definite covariance matrices. In the one-sample case, tests are proposed for sphericity and for testing the hypothesis that the covariance matrix Σ is an identity matrix, by providing an unbiased estimator of tr [ Σ 2 ] under the general model which requires no more computing time than the one available in the literature for a normal model. In the two-sample case, tests for the equality of two covariance matrices are given. The asymptotic distributions of proposed tests in the one-sample case are derived under the assumption that the sample size N = O ( p δ ) , 1 / 2 < δ < 1 , where p is the dimension of the random vector, and O ( p δ ) means that N / p goes to zero as N and p go to infinity. Similar assumptions are made in the two-sample case.