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 (2005) 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.

Final version accepted to Bernoulli, with some adjustments suitable for arXiv. Compared to the previous version, main changes include: a) A more general "impossibility result" is proved in Section 2. b) Extended discussion about open problems is included in Section 4 after the proof of Theorem 3.1 c) The proof of Theorem 3.2 is moved to Section 5