Cramér-type moderate deviations for Studentized two-sample $U$-statistics with applications
Jinyuan Chang, Qi-Man Shao, Wen-Xin Zhou
The Annals of Statistics · 2016 · Volume 44 · Issue 5
Abstract
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.
Keywords
60F10, 62E17, 62E20, 62F40, 62H15, 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
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