Approximations for the number of maxima and near-maxima in independent data

[2505.06088] Approximations for the number of maxima and near-maxima in independent data Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2505.06088 (math) [Submitted on 9 May 2025 ( v1 ), last revised 5 Mar 2026 (this version, v3)] Title: Approximations for the number of maxima and near-maxima in independent data Authors: Fraser Daly View a PDF of the paper titled Approximations for the number of maxima and near-maxima in independent data, by Fraser Daly View PDF HTML (experimental) Abstract: 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. Comments: 19 pages; 2 figures and 1 table. Minor edits in Example 7 Subjects: Probability (math.PR) ; Statistics Theory (math.ST) MSC classes: 62E17 (Primary) 60E05, 60E15 (Secondary) Cite as: arXiv:2505.06088 [math.PR] (or arXiv:2505.06088v3 [math.PR] for this version) https://doi.org/10.48550/arXiv.2505.06088 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Methodology and Computing in Applied Probabability 28: 33 (2026) Related DOI : https://doi.org/10.1007/s11009-026-10264-z Focus to learn more DOI(s) linking to related resources Submission history From: Fraser Daly [ view email ] [v1] Fri, 9 May 2025 14:32:17 UTC (23 KB) [v2] Wed, 26 Nov 2025 11:51:09 UTC (46 KB) [v3] Thu, 5 Mar 2026 09:21:49 UTC (46 KB) Full-text links: Access Paper: View a PDF of the paper titled Approximations for the number of maxima and near-maxima in independent data, by Fraser Daly View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2025-05 Change to browse by: math math.ST stat stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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