{"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/least-squares-estimation-of-two-ordered","title":"Least Squares estimation of two ordered monotone regression curves","arxiv_id":"0904.2052","date":"2009-04-14","proceeding":null,"authors":["Fadoua Balabdaoui","Kaspar Rufibach","Filippo Santambrogio"],"abstract":"In this paper, we consider the problem of finding the Least Squares estimators of two isotonic regression curves $g^\\circ_1$ and $g^\\circ_2$ under the additional constraint that they are ordered; e.g., $g^\\circ_1 \\le g^\\circ_2$. Given two sets of $n$ data points $y_1, ..., y_n$ and $z_1, >...,z_n$ observed at (the same) design points, the estimates of the true curves are obtained by minimizing the weighted Least Squares criterion $L_2(a, b) = \\sum_{j=1}^n (y_j - a_j)^2 w_{1,j}+ \\sum_{j=1}^n (z_j - b_j)^2 w_{2,j}$ over the class of pairs of vectors $(a, b) \\in \\mathbb{R}^n \\times \\mathbb{R}^n $ such that $a_1 \\le a_2 \\le ...\\le a_n $, $b_1 \\le b_2 \\le ...\\le b_n $, and $a_i \\le b_i, i=1, ...,n$. The characterization of the estimators is established. To compute these estimators, we use an iterative projected subgradient algorithm, where the projection is performed with a \"generalized\" pool-adjacent-violaters algorithm (PAVA), a byproduct of this work. Then, we apply the estimation method to real data from mechanical engineering.","url_abs":"https://arxiv.org/abs/0904.2052v3","url_pdf":"https://arxiv.org/pdf/0904.2052v3.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":"least-squares-estimation-of-two-ordered","repo_url":"https://cran.r-project.org/package=OrdMonReg","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}