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February 2021 Asymptotically independent U-statistics in high-dimensional testing
Yinqiu He, Gongjun Xu, Chong Wu, Wei Pan
Ann. Statist. 49(1): 154-181 (February 2021). DOI: 10.1214/20-AOS1951

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.

Citation

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Yinqiu He. Gongjun Xu. Chong Wu. Wei Pan. "Asymptotically independent U-statistics in high-dimensional testing." Ann. Statist. 49 (1) 154 - 181, February 2021. https://doi.org/10.1214/20-AOS1951

Information

Received: 1 May 2019; Revised: 1 October 2019; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/20-AOS1951

Subjects:
Primary: 62F03, 62F05

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 1 • February 2021
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