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 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.
翻译:我们根据第三步假设为1美元/\\sqrt{n}美元(B-E)的指数化指数建立了非统一的Berry-Esseen(B-E)界限,这一界限包括相当于一级核心的t-统计学,作为一个特例。虽然Novak(2005年)提出的一个有趣的数据实例可以表明,非统一的标准化U统计学约束形式对其学生化对应方实际上无效,但我们的主要结果表明,这种界限的有效性可以通过尽可能小地增加一个在n指数上指数化的添加词来恢复。 据我们所知,这是首次在文献中出现对学生化的U- Statistic的有效非统一的B-E界限。</s>