A Further Study on Chen-Qin's Test for Two-Sample Behrens-Fisher Problems for High-Dimensional Data

A further study on Chen-Qin's test, namely CQ test, for two-sample Behrens-Fisher problems for high-dimensional data is conducted, resulting in a new normal-reference test where the null distribution of the CQ-test statistic is approximated with that of a Chi-square-type mixture, which is obtained from the CQ-test statistic when the null hypothesis holds and when the two samples are normally distributed. The distribution of the Chi-square-type mixture can be well approximated by a three-cumulant matched chi(2) approximation with the approximation parameters consistently estimated from the data. The asymptotical power of the new normal-reference test under a local alternative is established. Two simulation studies demonstrate that in terms of size control, the new normal-reference test with the three-cumulant matched chi(2) -approximation performs well regardless of whether the data are nearly uncorrelated, moderately correlated, or highly correlated, and it performs substantially better than the CQ-test. A real data example illustrates the new normal-reference test.