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ASYMPTOTICALLY INDEPENDENT U-STATISTICS IN HIGH-DIMENSIONAL TESTING.
He, Yinqiu; Xu, Gongjun; Wu, Chong; Pan, Wei.
Affiliation
  • He Y; Department of Statistics, University of Michigan.
  • Xu G; Department of Statistics, University of Michigan.
  • Wu C; Department of Statistics, Florida State University.
  • Pan W; Division of Biostatistics, School of Public Health, University of Minnesota.
Ann Stat ; 49(1): 154-181, 2021 Feb.
Article in En | MEDLINE | ID: mdl-34857975
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.
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