Cramer type moderate deviations for Studentized two sample $U$ statistics with applications
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 Cramer 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.