Nonparametric Two-Sample Tests of High Dimensional Mean Vectors via Random Integration

Abstract

Testing the equality of the means in two samples is a fundamental statistical inferential problem. Most of the existing methods are based on the sum-of-squares or supremum statistics. They are possibly powerful in some situations, but not in others, and they do not work in a unified way. Using random integration of the difference, we develop a framework that includes and extends many existing methods, especially in high-dimensional settings, without restricting the same covariance matrices or sparsity. Under a general multivariate model, we can derive the asymptotic properties of the proposed test statistic without specifying a relationship between the data dimension and sample size explicitly. Specifically, the new framework allows us to better understand the test’s properties and select a powerful procedure accordingly. For example, we prove that our proposed test can achieve the power of 1 when nonzero signals in the true mean differences are weakly dense with nearly the same sign. In addition, we delineate the conditions under which the asymptotic relative Pitman efficiency of our proposed test to its competitor is greater than or equal to 1. Extensive numerical studies and a real data example demonstrate the potential of our proposed test. Supplementary materials for this article are available online.

Keywords:

• High-dimensional mean
• Nonparametric two-sample test
• Random integration of the difference

Supplementary Materials

We defer the technical proofs and details to the Supplementary Materials. Additional simulation results and analysis of the real dataset are also presented in the Supplementary Materials.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Jiang’s research is partially supported by NSFC (12171203) and the Natural Science Foundation of Guangdong (2022A1515010045). Wen’s research is partially supported by National Science Foundation of China (12171449). Wang’s research is partially supported by NSFC (12231017, 72171216, 71921001, 71991474), and the National Key R&D Program of China (2022YFA1003803). Zhang’s research is partially supported by the U.S. National Institutes of Health (R01HG010171 and R01MH116527) and NSF (DMS-2112711).