{"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/limit-distribution-theory-for-maximum","title":"Limit distribution theory for maximum likelihood estimation of a log-concave density","arxiv_id":"0708.3400","date":"2007-08-24","proceeding":null,"authors":["Fadoua Balabdaoui","Kaspar Rufibach","Jon A. Wellner"],"abstract":"We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\\exp\\varphi_0$ where $\\varphi_0$ is a concave function on $\\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the \"lower invelope\" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\\varphi_0=\\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.","url_abs":"https://arxiv.org/abs/0708.3400v3","url_pdf":"https://arxiv.org/pdf/0708.3400v3.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":"limit-distribution-theory-for-maximum","repo_url":"https://cran.r-project.org/package=logcondens","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"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}