Finite-sample properties of the trimmed mean
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915093494628352 |
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| author | Oliveira, Roberto I. Orenstein, Paulo Rico, Zoraida F. |
| author_facet | Oliveira, Roberto I. Orenstein, Paulo Rico, Zoraida F. |
| contents | The trimmed mean of $n$ scalar random variables from a distribution $P$ is the variant of the standard sample mean where the $k$ smallest and $k$ largest values in the sample are discarded for some parameter $k$. In this paper, we look at the finite-sample properties of the trimmed mean as an estimator for the mean of $P$. Assuming finite variance, we prove that the trimmed mean is ``sub-Gaussian'' in the sense of achieving Gaussian-type concentration around the mean. Under slightly stronger assumptions, we show the left and right tails of the trimmed mean satisfy a strong ratio-type approximation by the corresponding Gaussian tail, even for very small probabilities of the order $e^{-n^c}$ for some $c>0$. In the more challenging setting of weaker moment assumptions and adversarial sample contamination, we prove that the trimmed mean is minimax-optimal up to constants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite-sample properties of the trimmed mean Oliveira, Roberto I. Orenstein, Paulo Rico, Zoraida F. Statistics Theory Probability 62C20 (Primary) 60F10 (Secondary) The trimmed mean of $n$ scalar random variables from a distribution $P$ is the variant of the standard sample mean where the $k$ smallest and $k$ largest values in the sample are discarded for some parameter $k$. In this paper, we look at the finite-sample properties of the trimmed mean as an estimator for the mean of $P$. Assuming finite variance, we prove that the trimmed mean is ``sub-Gaussian'' in the sense of achieving Gaussian-type concentration around the mean. Under slightly stronger assumptions, we show the left and right tails of the trimmed mean satisfy a strong ratio-type approximation by the corresponding Gaussian tail, even for very small probabilities of the order $e^{-n^c}$ for some $c>0$. In the more challenging setting of weaker moment assumptions and adversarial sample contamination, we prove that the trimmed mean is minimax-optimal up to constants. |
| title | Finite-sample properties of the trimmed mean |
| topic | Statistics Theory Probability 62C20 (Primary) 60F10 (Secondary) |
| url | https://arxiv.org/abs/2501.03694 |