Nonuniform Berry-Esseen bounds for studentized U-statistics

Abstract

We establish nonuniform Berry-Esseen (B-E) bounds for Studentized U-statistics of the rate $1∕\sqrt{n}$ under a third-moment assumption, which covers the t-statistic that corresponds to a kernel of degree 1 as a special case. While an interesting data example raised by Novak (Theory Probab. Appl. 49 (2005) 336–344) can show that the form of the nonuniform bound for standardized U-statistics is actually invalid for their Studentized counterparts, our main results suggest that, the validity of such a bound can be restored by minimally augmenting it with an additive correction term that decays exponentially in n. To our best knowledge, this is the first time that valid nonuniform B-E bounds for Studentized U-statistics have appeared in the literature.