Cram\'{e}r-type moderate deviations for Studentized two-sample $U$-statistics with applications

TL;DR

This paper develops sharp Cramér-type moderate deviation theorems for Studentized two-sample U-statistics, including the two-sample t-statistic, enhancing the understanding of their tail behaviors and applications in multiple testing.

Contribution

It establishes the first refined moderate deviation theorem with second-order accuracy for Studentized two-sample U-statistics, extending existing results to more general nonlinear statistics.

Findings

Derived sharp Cramér-type moderate deviation theorems for Studentized two-sample U-statistics.

Extended applicability of statistical methodologies from one-sample to two-sample nonlinear statistics.

Discussed applications in large-scale multiple testing and bootstrap methods.

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\'{e}r-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.