He Yinqiu, Xu Gongjun, Wu Chong, Pan Wei
Department of Statistics, University of Michigan.
Department of Statistics, Florida State University.
Ann Stat. 2021 Feb;49(1):154-181. doi: 10.1214/20-aos1951. Epub 2021 Jan 29.
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 -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 -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.
许多高维假设检验旨在全局检验高维联合分布的边际或低维特征，例如均值向量、协方差矩阵和回归系数的检验。本文构造了一族U统计量作为这些特征的 -范数的无偏估计量。我们表明，在原假设下，不同有限阶的U统计量渐近独立且服从正态分布。此外，它们与最大型检验统计量也渐近独立，其极限分布为极值分布。基于渐近独立性性质，我们提出了一种自适应检验程序，该程序结合了从不同阶的U统计量计算出的 -值。我们进一步建立了功效分析结果，并表明所提出的自适应程序在面对各种备择假设时都保持高功效。
