Standardized Dempster's Non-Exact Test for High-Dimensional Mean Vectors

Although the Hotelling's (Formula presented.) test has been a widely used test for hypothesis testing problems on the mean vectors, it is not well defined when the data dimension is larger than the sample size. Dempster's non-exact test, as a remedy for the Hotelling's (Formula presented.) test, is known to be more powerful than the Hotelling's (Formula presented.) test and is well defined even when the dimension is much larger than the sample size. However, Dempster's non-exact test will lose power when the variances of the covariates are different. In this paper, we propose a standardized Dempster's non-exact test for the classical mean testing problem. The proposed test is more powerful for data with heteroscedastic features and is applicable to the high-dimensional case. An approximate distribution of the test statistic has been established, and to better control the type I error rate when the sample size is small, we further constructed a Monte Carlo version of the proposed standardized Dempster's non-exact test. Various simulation studies and a real data application were conducted with comparison to other popular tests. The numerical results showed that while the type I error rates were well controlled, the testing power of our proposed test was generally higher than those of other tests.