ASYMPTOTICALLY INDEPENDENT U-STATISTICS IN HIGH-DIMENSIONAL TESTING

• 成果类型: Article
• 署名作者: He, Yinqiu; Xu, Gongjun; Wu, Chong; Pan, Wei
• 署名单位: University of Michigan System; University of Michigan; State University System of Florida; Florida State University; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1951
• 发表日期: 2021
• 页码: 154-181
• 关键词: covariance-matrix; HIGHER CRITICISM; 2-sample test; LARGEST EIGENVALUES; fewer observations; maximum; SUM; distributions; association; UNIVERSALITY

摘要

Many high-dimensional hypothesis tests aim to globally examine marginal or low-dimensional features of a high-dimensional joint distribution, such as testing of mean vectors, covariance matrices and regression coefficients. This paper constructs a family of U-statistics as unbiased estimators of the l(p)-norms of those features. We show that under the null hypothesis, the U-statistics of different finite orders are asymptotically independent and normally distributed. Moreover, they are also asymptotically independent with the maximum-type test statistic, whose limiting distribution is an extreme value distribution. Based on the asymptotic independence property, we propose an adaptive testing procedure which combines p-values computed from the U-statistics of different orders. We further establish power analysis results and show that the proposed adaptive procedure maintains high power against various alternatives.