  We propose a multiple-splitting projection test (MPT) for one-sample mean
vectors in high-dimensional settings. The idea of projection test is to project
high-dimensional samples to a 1-dimensional space using an optimal projection
direction such that traditional tests can be carried out with projected
samples. However, estimation of the optimal projection direction has not been
systematically studied in the literature. In this work, we bridge the gap by
proposing a consistent estimation via regularized quadratic optimization. To
retain type I error rate, we adopt a data-splitting strategy when constructing
test statistics. To mitigate the power loss due to data-splitting, we further
propose a test via multiple splits to enhance the testing power. We show that
the $p$-values resulted from multiple splits are exchangeable. Unlike existing
methods which tend to conservatively combine dependent $p$-values, we develop
an exact level $\alpha$ test that explicitly utilizes the exchangeability
structure to achieve better power. Numerical studies show that the proposed
test well retains the type I error rate and is more powerful than
state-of-the-art tests.
