Nonuniform Berry-Esseen bounds for Studentized U-statistics
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917576422981632 |
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| author | Leung, Dennis Shao, Qi-Man |
| author_facet | Leung, Dennis Shao, Qi-Man |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08619 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonuniform Berry-Esseen bounds for Studentized U-statistics Leung, Dennis Shao, Qi-Man Statistics Theory 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. |
| title | Nonuniform Berry-Esseen bounds for Studentized U-statistics |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2303.08619 |