Projection Test for Mean Vector in High Dimensions

Wanjun Liu, Xiufan Yu, Wei Zhong and Runze Li

Journal of the American Statistical Association, 2024, vol. 119, issue 545, 744-756

Abstract: 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 t-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.

Date: 2024
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DOI: 10.1080/01621459.2022.2142592

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