Projection Test for Mean Vector in High Dimensions

This article studies the projection test for high-dimensional mean vectors via optimal projection. The idea of projection test is to project high-dimensional data onto a space of low dimension such that traditional methods can be applied. We first propose a new estimation for the optimal projection direction by solving a constrained and regularized quadratic programming. Then two tests are constructed using the estimated optimal projection direction. The first one is based on a data-splitting procedure, which achieves an exact <i>t</i>-test under normality assumption. To mitigate the power loss due to data-splitting, we further propose an online framework, which iteratively updates the estimation of projection direction when new observations arrive. We show that this online-style projection test asymptotically converges to the standard normal distribution. Various simulation studies as well as a real data example show that the proposed online-style projection test retains the Type I error rate well and is more powerful than other existing tests. Supplementary materials for this article are available online.