The Annals of Statistics

Randomized incomplete $U$-statistics in high dimensions

Xiaohui Chen and Kengo Kato

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This paper studies inference for the mean vector of a high-dimensional $U$-statistic. In the era of big data, the dimension $d$ of the $U$-statistic and the sample size $n$ of the observations tend to be both large, and the computation of the $U$-statistic is prohibitively demanding. Data-dependent inferential procedures such as the empirical bootstrap for $U$-statistics is even more computationally expensive. To overcome such a computational bottleneck, incomplete $U$-statistics obtained by sampling fewer terms of the $U$-statistic are attractive alternatives. In this paper, we introduce randomized incomplete $U$-statistics with sparse weights whose computational cost can be made independent of the order of the $U$-statistic. We derive nonasymptotic Gaussian approximation error bounds for the randomized incomplete $U$-statistics in high dimensions, namely in cases where the dimension $d$ is possibly much larger than the sample size $n$, for both nondegenerate and degenerate kernels. In addition, we propose generic bootstrap methods for the incomplete $U$-statistics that are computationally much less demanding than existing bootstrap methods, and establish finite sample validity of the proposed bootstrap methods. Our methods are illustrated on the application to nonparametric testing for the pairwise independence of a high-dimensional random vector under weaker assumptions than those appearing in the literature.

Article information

Ann. Statist., Volume 47, Number 6 (2019), 3127-3156.

Received: December 2017
Revised: October 2018
First available in Project Euclid: 31 October 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 62E17: Approximations to distributions (nonasymptotic) 62F40: Bootstrap, jackknife and other resampling methods
Secondary: 62H15: Hypothesis testing

Incomplete $U$-statistics randomized inference Gaussian approximation bootstrap divide and conquer Bernoulli sampling sampling with replacement


Chen, Xiaohui; Kato, Kengo. Randomized incomplete $U$-statistics in high dimensions. Ann. Statist. 47 (2019), no. 6, 3127--3156. doi:10.1214/18-AOS1773.

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Supplemental materials

  • Supplement to “Randomized incomplete $U$-statistics in high dimensions”. The Supplementary Material contains the proofs and additional discussions, simulation results and applications of the main paper.