The Annals of Statistics

Cramér-type moderate deviations for Studentized two-sample $U$-statistics with applications

Jinyuan Chang, Qi-Man Shao, and Wen-Xin Zhou

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Two-sample $U$-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér-type moderate deviation theorems for Studentized two-sample $U$-statistics in a general framework, including the two-sample $t$-statistic and Studentized Mann–Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample $t$-statistic. These results extend the applicability of the existing statistical methodologies from the one-sample $t$-statistic to more general nonlinear statistics. Applications to two-sample large-scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed.

Article information

Ann. Statist., Volume 44, Number 5 (2016), 1931-1956.

Received: June 2015
First available in Project Euclid: 12 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 62E17: Approximations to distributions (nonasymptotic)
Secondary: 62E20: Asymptotic distribution theory 62F40: Bootstrap, jackknife and other resampling methods 62H15: Hypothesis testing

Bootstrap false discovery rate Mann–Whitney $U$ test multiple hypothesis testing self-normalized moderate deviation Studentized statistics two-sample $t$-statistic two-sample $U$-statistics


Chang, Jinyuan; Shao, Qi-Man; Zhou, Wen-Xin. Cramér-type moderate deviations for Studentized two-sample $U$-statistics with applications. Ann. Statist. 44 (2016), no. 5, 1931--1956. doi:10.1214/15-AOS1375.

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Supplemental materials

  • Supplement to “Cramér-type moderate deviations for Studentized two-sample $U$-statistics with applications”. This supplemental material contains proofs for all the theoretical results in the main text, including Theorems 2.2, 2.4, 3.1 and 3.4, and additional numerical results.