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# Cram\'er type moderate deviations for two-sample Studentized $$U$$-statistics with applications

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### 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 Cram\'er 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.

### Author and article information

###### Journal
2014-07-16
2015-11-13
10.1214/15-AOS1375
1407.4546