Asymptotically independent U-statistics in high-dimensional testing
本文构造了一族U统计量作为高维分布特征的ℓp范数的无偏估计，证明了不同阶U统计量在零假设下渐近独立且正态分布，并基于此提出自适应检验程序，结合不同阶的p值以应对多种备择假设。
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 $\ell_{p}$-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.