A revisit to Bai–Saranadasa's two-sample test

Bai–Saranadasa's two-sample test for high-dimensional data, namely BS-test, has been widely cited in the literature. However, it may not control the size well when the required conditions are not satisfied. In this paper, a revisit to the BS-test is conducted. It is shown that under some regularity conditions and the null hypothesis, the BS-test statistic and a chi-square-type mixture have the same limiting distribution. It is then natural to approximate the null distribution of the BS-test using that of the chi-square-type mixture, which is actually obtained from the BS-test statistic when the two high-dimensional samples are normally distributed. The resulting test is then referred to as a normal-reference test. Two simulation studies and a real data example demonstrate that in terms of size control, the proposed normal-reference test performs very well and it performs substantially better than the BS-test and three other existing competitors proposed in the literature.