A Simple Two-Sample Test in High Dimensions Based on L2-Norm

• 成果类型: Article
• 署名作者: Zhang, Jin-Ting; Guo, Jia; Zhou, Bu; Cheng, Ming-Yen
• 署名单位: National University of Singapore; Zhejiang University of Technology; Zhejiang Gongshang University; Hong Kong Baptist University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2019.1604366
• 发表日期: 2020
• 页码: 1011-1027
• 关键词: nonconcave penalized likelihood; asymptotic distributions; fewer observations; COVARIANCE-MATRIX; mean vector; CLASSIFICATION; selection; regression; reduction; MODEL

摘要

Testing the equality of two means is a fundamental inference problem. For high-dimensional data, the Hotelling's T-2-test either performs poorly or becomes inapplicable. Several modifications have been proposed to address this issue. However, most of them are based on asymptotic normality of the null distributions of their test statistics which inevitably requires strong assumptions on the covariance. We study this problem thoroughly and propose an L-2-norm based test that works under mild conditions and even when there are fewer observations than the dimension. Specially, to cope with general nonnormality of the null distribution we employ the Welch-Satterthwaite chi(2)-approximation. We derive a sharp upper bound on the approximation error and use it to justify that chi(2)-approximation is preferred to normal approximation. Simple ratio-consistent estimators for the parameters in the chi(2)-approximation are given. Importantly, our test can cope with singularity or near singularity of the covariance which is commonly seen in high dimensions and is the main cause of nonnormality. The power of the proposed test is also investigated. Extensive simulation studies and an application show that our test is at least comparable to and often outperforms several competitors in terms of size control, and the powers are comparable when their sizes are. for this article are available online.