Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866908469624307712 |
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| author | Derumigny, Alexis Girard, Lucas Guyonvarch, Yannick |
| author_facet | Derumigny, Alexis Girard, Lucas Guyonvarch, Yannick |
| contents | In this article, we obtain explicit bounds on the uniform distance between the cumulative distribution function of a standardized sum $S_n$ of $n$ independent centered random variables with moments of order four and its first-order Edgeworth expansion. Those bounds are valid for any sample size with $n^{-1/2}$ rate under moment conditions only and $n^{-1}$ rate under additional regularity constraints on the tail behavior of the characteristic function of $S_n$. In both cases, the bounds are further sharpened if the variables involved in $S_n$ are unskewed. We also derive new Berry-Esseen-type bounds from our results and discuss their links with existing ones. We finally apply our results to illustrate the lack of finite-sample validity of one-sided tests based on the normal approximation of the mean. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_05780 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion Derumigny, Alexis Girard, Lucas Guyonvarch, Yannick Probability Econometrics Statistics Theory Primary: 62E17, Secondary: 60F05, 62F03 In this article, we obtain explicit bounds on the uniform distance between the cumulative distribution function of a standardized sum $S_n$ of $n$ independent centered random variables with moments of order four and its first-order Edgeworth expansion. Those bounds are valid for any sample size with $n^{-1/2}$ rate under moment conditions only and $n^{-1}$ rate under additional regularity constraints on the tail behavior of the characteristic function of $S_n$. In both cases, the bounds are further sharpened if the variables involved in $S_n$ are unskewed. We also derive new Berry-Esseen-type bounds from our results and discuss their links with existing ones. We finally apply our results to illustrate the lack of finite-sample validity of one-sided tests based on the normal approximation of the mean. |
| title | Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion |
| topic | Probability Econometrics Statistics Theory Primary: 62E17, Secondary: 60F05, 62F03 |
| url | https://arxiv.org/abs/2101.05780 |