{"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/profile-least-squares-estimators-in-the","title":"Profile least squares estimators in the monotone single index model","arxiv_id":"2001.05454","date":"2020-01-15","proceeding":null,"authors":["Fadoua Balabdaoui","Piet Groeneboom"],"abstract":"We consider least squares estimators of the finite regression parameter $\\alpha$ in the single index regression model $Y=\\psi(\\alpha^T X)+\\epsilon$, where $X$ is a $d$-dimensional random vector, $\\E(Y|X)=\\psi(\\alpha^T X)$, and where $\\psi$ is monotone. It has been suggested to estimate $\\alpha$ by a profile least squares estimator, minimizing $\\sum_{i=1}^n(Y_i-\\psi(\\alpha^T X_i))^2$ over monotone $\\psi$ and $\\alpha$ on the boundary $S_{d-1}$of the unit ball. Although this suggestion has been around for a long time, it is still unknown whether the estimate is $\\sqrt{n}$ convergent. We show that a profile least squares estimator, using the same pointwise least squares estimator for fixed $\\alpha$, but using a different global sum of squares, is $\\sqrt{n}$-convergent and asymptotically normal. The difference between the corresponding loss functions is studied and also a comparison with other methods is given.","url_abs":"https://arxiv.org/abs/2001.05454v3","url_pdf":"https://arxiv.org/pdf/2001.05454v3.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":"profile-least-squares-estimators-in-the","repo_url":"https://github.com/pietg/single_index","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}