Asymptotically independent U-statistics in high-dimensional testing

2021/01/29 by Yinqiu He, Gongjun Xu, Chong Wu +1 · 50 citations
Mathematics · #Applied mathematics #Asymptotic distribution #Covariance #Efficiency #Estimator #Independence (probability theory) #Mathematics #Null distribution #Order statistic #Random Matrices and Applications #Statistic #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistical hypothesis testing #Statistics #Test statistic #U-statistic #p-value

paper · open access · doi:10.1214/20-aos1951

published in The Annals of Statistics 49(1), 154-181 (Institute of Mathematical Statistics)

openalex publication_date 2021/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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 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.

Citations

• On the distribution of the largest eigenvalue in principal components analysis
• Statistical significance for genomewide studies
• A test for the mean vector with fewer observations than the dimension
• Power Enhancement in High-Dimensional Cross-Sectional Tests

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