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

Srivastava, Muni S., Hirokazu Yanagihara and Tatsuya Kubokawa

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 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.}