Nonuniform Berry-Esseen bounds for Studentized U-statistics

TL;DR

This paper derives the first valid nonuniform Berry-Esseen bounds for Studentized U-statistics, showing that with a small correction, these bounds accurately describe their distributional convergence.

Contribution

It introduces the first valid nonuniform Berry-Esseen bounds for Studentized U-statistics, incorporating an exponential decay correction term.

Findings

Bounds are valid under a third-moment condition.

The correction term decays exponentially with sample size.

The results include the t-statistic as a special case.

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.