Two-sample Behrens–Fisher problems for high-dimensional data: a normal reference scale-invariant test

For high-dimensional two-sample Behrens–Fisher problems, several non-scale-invariant and scale-invariant tests have been proposed. Most of them impose strong assumptions on the underlying group covariance matrices so that their test statistics are asymptotically normal. However, in practice, these assumptions may not be satisfied or hardly be checked so that these tests may not be able to maintain the nominal size well in practice. To overcome this difficulty, in this paper, a normal reference scale-invariant test is proposed and studied. It works well by neither imposing strong assumptions on the underlying group covariance matrices nor assuming their equality. It is shown that under some regularity conditions and the null hypothesis, the proposed test and a chi-square-type mixture have the same normal and non-normal limiting distributions. It is then justifiable to approximate the null distribution of the proposed test using that of the chi-square-type mixture. The distribution of the chi-square type mixture can be well approximated by the Welch–Satterthwaite chi-square-approximation with the approximation parameter consistently estimated from the data. The asymptotic power of the proposed test is established. Numerical results demonstrate that the proposed test has much better size control and power than several well-known non-scale-invariant and scale-invariant tests.