Testing the equality of several covariance matrices with fewer observations than the dimension

For normally distributed data from the kk populations with m×mm×m covariance matrices Σ1,…,ΣkΣ1,…,Σk, we test the hypothesis H:Σ1=⋯=ΣkH:Σ1=⋯=Σk vs the alternative A≠HA≠H when the number of observations NiNi, i=1,…,ki=1,…,k from each population are less than or equal to the dimension mm, Ni≤mNi≤m, i=1,…,ki=1,…,k. Two tests are proposed and compared with two other tests proposed in the literature. These tests, however, do not require that Ni≤mNi≤m, and thus can be used in all situations, including when the likelihood ratio test is available. The asymptotic distributions of the test statistics are given, and the power compared by simulations with other test statistics proposed in the literature. The proposed tests perform well and better in several cases than the other two tests available in the literature.