Asymptotically independent U-statistics in high-dimensional testing
Yinqiu He, Gongjun Xu, Chong Wu, Wei Pan
The Annals of Statistics · 2021 · Volume 49 · Issue 1
Abstract
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.
Keywords
62F03, 62F05, Adaptive testing, High-dimensional hypothesis test, U-statistics
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