"Tests for Covariance Matrices in High Dimension with Less Sample Size"

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