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arXiv · 2303.08619

Nonuniform Berry-Esseen bounds for Studentized U-statistics

Dennis Leung·Qi-Man Shao
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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.

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Dennis Leung, Qi-Man Shao. 2023-03-15. Nonuniform Berry-Esseen bounds for Studentized U-statistics. https://arxiv.org/abs/2303.08619

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