  Size distortion can occur if an asymptotic testing procedure requiring
diverging sample sizes, is implemented to data with very small sample sizes. In
this paper, we consider one-sample and two-sample tests for mean vectors when
data are high-dimensional but sample sizes are very small. We establish
asymptotic t-distributions of one-sample and two-sample U-statistics, which
only require data dimensionality to diverge but sample sizes to be fixed and no
less than 3. Simulation studies confirm the theoretical results that the
proposed tests maintain accurate empirical sizes for a wide range of sample
sizes and data dimensionalities. We apply the proposed tests to an fMRI dataset
to demonstrate the practical implementation of the methods.
