{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/maximum-likelihood-fits-of-piece-wise-pareto","title":"Maximum-likelihood fits of piece-wise Pareto distributions with finite and non-zero core","arxiv_id":"2309.09589","date":"2023-09-18","proceeding":null,"authors":["Benjamin F. Maier"],"abstract":"We discuss multiple classes of piece-wise Pareto-like power law probability density functions $p(x)$ with two regimes, a non-pathological core with non-zero, finite values for support $0\\leq x\\leq x_{\\mathrm{min}}$ and a power-law tail with exponent $-\\alpha$ for $x>x_{\\mathrm{min}}$. The cores take the respective shapes (i) $p(x)\\propto (x/x_{\\mathrm{min}})^\\beta$, (ii) $p(x)\\propto\\exp(-\\beta[x/x_{\\mathrm{min}}-1])$, and (iii) $p(x)\\propto [2-(x/x_{\\mathrm{min}})^\\beta]$, including the special case $\\beta=0$ leading to core $p(x)=\\mathrm{const}$. We derive explicit maximum-likelihood estimators and/or efficient numerical methods to find the best-fit parameter values for empirical data. Solutions for the special cases $\\alpha=\\beta$ are presented, as well. The results are made available as a Python package.","url_abs":"https://arxiv.org/abs/2309.09589v1","url_pdf":"https://arxiv.org/pdf/2309.09589v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"maximum-likelihood-fits-of-piece-wise-pareto","repo_url":"https://github.com/benmaier/fincoretails","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}