Testing the structure of the covariance matrix with fewer observations than the dimension

We consider two hypothesis testing problems with NN independent observations on a single mm-vector, when m>Nm>N, and the NN observations on the random mm-vector are independently and identically distributed as multivariate normal with mean vector μ and covariance matrix ΣΣ, both unknown. In the first problem, the mm-vector is partitioned into two sub-vectors of dimensions m1m1 and m2m2, respectively, and we propose two tests for the independence of the two sub-vectors that are valid as (m,N)→∞(m,N)→∞. The asymptotic distribution of the test statistics under the hypothesis of independence is shown to be standard normal, and the power examined by simulations. The proposed tests perform better than the likelihood ratio test, although the latter can only be used when mm is smaller than NN. The second problem addressed is that of testing the hypothesis that the covariance matrix ΣΣ is of the intraclass correlation structure. A statistic for testing this is proposed, and assessed via simulations; again the proposed test statistic compares favorably with the likelihood ratio test.