Approximations for the number of maxima and near-maxima in independent data
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913017059344384 |
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| author | Daly, Fraser |
| author_facet | Daly, Fraser |
| contents | In the setting where we have $n$ independent observations of a random variable $X$, we derive explicit error bounds in total variation distance when approximating the number of observations equal to the maximum of the sample (in the case where $X$ is discrete) or the number of observations within a given distance of an order statistic of the sample (in the case where $X$ is absolutely continuous). The logarithmic and Poisson distributions are used as approximations in the discrete case, with proofs which include the development of Stein's method for a logarithmic target distribution. In the absolutely continuous case our approximations are by the negative binomial distribution, and are established by considering negative binomial approximation for mixed binomials. The cases where $X$ is geometric, Gumbel and uniform are used as illustrative examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximations for the number of maxima and near-maxima in independent data Daly, Fraser Probability Statistics Theory 62E17 (Primary) 60E05, 60E15 (Secondary) In the setting where we have $n$ independent observations of a random variable $X$, we derive explicit error bounds in total variation distance when approximating the number of observations equal to the maximum of the sample (in the case where $X$ is discrete) or the number of observations within a given distance of an order statistic of the sample (in the case where $X$ is absolutely continuous). The logarithmic and Poisson distributions are used as approximations in the discrete case, with proofs which include the development of Stein's method for a logarithmic target distribution. In the absolutely continuous case our approximations are by the negative binomial distribution, and are established by considering negative binomial approximation for mixed binomials. The cases where $X$ is geometric, Gumbel and uniform are used as illustrative examples. |
| title | Approximations for the number of maxima and near-maxima in independent data |
| topic | Probability Statistics Theory 62E17 (Primary) 60E05, 60E15 (Secondary) |
| url | https://arxiv.org/abs/2505.06088 |