Finite sample t-tests for high-dimensional means.

When sample sizes are small, it becomes challenging for an asymptotic test requiring diverging sample sizes to maintain an accurate Type I error rate. In this paper, we consider one-sample, two-sample and ANOVA tests for mean vectors when data are high-dimensional but sample sizes are very small. We establish asymptotic t-distributions of the proposed U-statistics, which only require data dimensionality to diverge but sample sizes to be fixed and no less than 3. The proposed tests maintain accurate Type I error rates for a wide range of sample sizes and data dimensionality. Moreover, the tests are nonparametric and can be applied to data which are normally distributed or heavy-tailed. Simulation studies confirm the theoretical results for the tests. We also apply the proposed tests to an fMRI dataset to demonstrate the practical implementation of the methods.