{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T11:02:53Z","timestamp":1789556573153,"version":"build-2803163510"},"reference-count":50,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:00:00Z","timestamp":1770681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271510"],"award-info":[{"award-number":["12271510"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,9,16]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Empirical likelihood encounters serious computational challenges when applied to massive datasets or multiple data sources distributed across decentralized networks. This paper proposes a constrained empirical likelihood framework for decentralized networks, utilizing a novel penalization technique to obtain a penalized empirical log-likelihood. The resulting empirical log-likelihood ratio statistic is proved to be asymptotically standard chi-squared even for a divergent machine number. However, the optimization problem with the fused penalty is still hard to solve in the decentralized distributed network due to the coupling structure. To address the problem, two novel algorithms are developed to solve the optimization problem in a decentralized manner, with established convergence properties and linear convergence for the second algorithm in specific network structures. The methods are validated through simulations and real data analyses of census income and Ford gobike datasets.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkag042","type":"journal-article","created":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T12:29:16Z","timestamp":1768652956000},"page":"1303-1320","source":"Crossref","is-referenced-by-count":1,"title":["Penalized empirical likelihood over decentralized networks"],"prefix":"10.1093","volume":"88","author":[{"given":"Jinye","family":"Du","sequence":"first","affiliation":[{"name":"SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences , Beijing 100190 ,","place":["China"]},{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences , Beijing 100049 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3708-8504","authenticated-orcid":false,"given":"Qihua","family":"Wang","sequence":"additional","affiliation":[{"name":"SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences , Beijing 100190 ,","place":["China"]},{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences , Beijing 100049 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,2,10]]},"reference":[{"issue":"3","key":"2026091606124096000_qkag042-B1","doi-asserted-by":"publisher","first-page":"1352","DOI":"10.1214\/17-AOS1587","article-title":"Distributed testing and estimation under sparse high dimensional models","volume":"46","author":"Battey","year":"2018","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026091606124096000_qkag042-B2","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1093\/biomet\/asaa051","article-title":"High-dimensional empirical likelihood inference","volume":"108","author":"Chang","year":"2021","journal-title":"Biometrika"},{"issue":"3","key":"2026091606124096000_qkag042-B3","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/s11749-009-0159-5","article-title":"A review on empirical likelihood methods for regression","volume":"18","author":"Chen","year":"2009","journal-title":"Test"},{"issue":"540","key":"2026091606124096000_qkag042-B4","doi-asserted-by":"publisher","first-page":"1775","DOI":"10.1080\/01621459.2021.1886937","article-title":"Distributed estimation for principal component analysis: An enlarged eigenspace analysis","volume":"117","author":"Chen","year":"2022","journal-title":"Journal of the American Statistical Association"},{"issue":"6","key":"2026091606124096000_qkag042-B5","doi-asserted-by":"publisher","first-page":"3244","DOI":"10.1214\/18-AOS1777","article-title":"Quantile regression under memory constraint","volume":"47","author":"Chen","year":"2019","journal-title":"Annals of Statistics"},{"key":"2026091606124096000_qkag042-B6","author":"Chow","year":"2016"},{"issue":"3","key":"2026091606124096000_qkag042-B7","doi-asserted-by":"publisher","first-page":"889","DOI":"10.1007\/s10915-015-0048-x","article-title":"On the global and linear convergence of the generalized alternating direction method of multipliers","volume":"66","author":"Deng","year":"2016","journal-title":"Journal of Scientific Computing"},{"issue":"3","key":"2026091606124096000_qkag042-B8","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1109\/TAC.2011.2161027","article-title":"Dual averaging for distributed optimization: Convergence analysis and network scaling","volume":"57","author":"Duchi","year":"2011","journal-title":"IEEE Transactions on Automatic Control"},{"key":"2026091606124096000_qkag042-B9","doi-asserted-by":"publisher","first-page":"290","DOI":"10.5486\/pmd.1959.6.3-4.12","article-title":"On random graphs. I","volume":"6","author":"Erd\u0151s","year":"1959","journal-title":"Publicationes Mathematicae Debrecen"},{"key":"2026091606124096000_qkag042-B10","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1090\/S0002-9947-1984-0756039-5","article-title":"On the evolution of random graphs","volume":"5","author":"Erd\u0151s","year":"1960","journal-title":"A Magyar Tudom\u00e1nyos Akad\u00e9mia. Matematikai Kutat\u00f3 Int\u00e9zet\u00e9nek K\u00f6zlem\u00e9nyei"},{"key":"2026091606124096000_qkag042-B11","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/j.jmva.2016.01.009","article-title":"Penalized empirical likelihood for high-dimensional partially linear varying coefficient model with measurement errors","volume":"147","author":"Fan","year":"2016","journal-title":"Journal of Multivariate Analysis"},{"issue":"456","key":"2026091606124096000_qkag042-B12","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1198\/016214501753382273","article-title":"Variable selection via nonconcave penalized likelihood and its oracle properties","volume":"96","author":"Fan","year":"2001","journal-title":"Journal of the American Statistical Association"},{"issue":"6","key":"2026091606124096000_qkag042-B13","doi-asserted-by":"publisher","first-page":"3009","DOI":"10.1214\/18-AOS1713","article-title":"Distributed estimation of principal eigenspaces","volume":"47","author":"Fan","year":"2019","journal-title":"Annals of Statistics"},{"issue":"2","key":"2026091606124096000_qkag042-B14","doi-asserted-by":"publisher","first-page":"109","DOI":"10.2307\/1403462","article-title":"Methodology and algorithms of empirical likelihood","volume":"5","author":"Hall","year":"1990","journal-title":"International Statistical Review = Revue Internationale De Statistique"},{"key":"2026091606124096000_qkag042-B15","author":"Hallac","year":"2015"},{"issue":"514","key":"2026091606124096000_qkag042-B16","doi-asserted-by":"publisher","first-page":"646","DOI":"10.1080\/01621459.2015.1024058","article-title":"Empirical likelihood for right censored lifetime data","volume":"111","author":"He","year":"2016","journal-title":"Journal of the American Statistical Association"},{"issue":"534","key":"2026091606124096000_qkag042-B17","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1080\/01621459.2020.1736082","article-title":"A distributed and integrated method of moments for high-dimensional correlated data analysis","volume":"116","author":"Hector","year":"2021","journal-title":"Journal of the American Statistical Association"},{"issue":"526","key":"2026091606124096000_qkag042-B18","doi-asserted-by":"publisher","first-page":"668","DOI":"10.1080\/01621459.2018.1429274","article-title":"Communication-efficient distributed statistical inference","volume":"114","author":"Jordan","year":"2019","journal-title":"Journal of the American Statistical Association"},{"issue":"509","key":"2026091606124096000_qkag042-B19","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1080\/01621459.2014.892882","article-title":"Homogeneity pursuit","volume":"110","author":"Ke","year":"2015","journal-title":"Journal of the American Statistical Association"},{"key":"2026091606124096000_qkag042-B20","doi-asserted-by":"publisher","DOI":"10.24432\/C5GP7S","volume-title":"Census income","author":"Kohavi","year":"1996"},{"issue":"1","key":"2026091606124096000_qkag042-B21","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1146\/statistics.2021.8.issue-1","article-title":"A review of empirical likelihood","volume":"8","author":"Lazar","year":"2021","journal-title":"Annual Review of Statistics and Its Application"},{"issue":"3","key":"2026091606124096000_qkag042-B22","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1093\/biomet\/ass014","article-title":"Penalized empirical likelihood and growing dimensional general estimating equations","volume":"99","author":"Leng","year":"2012","journal-title":"Biometrika"},{"issue":"182","key":"2026091606124096000_qkag042-B23","first-page":"1","article-title":"Divide-and-conquer for debiased l1-norm support vector machine in ultra-high dimensions","volume":"18","author":"Lian","year":"2018","journal-title":"Journal of Machine Learning Research"},{"issue":"517","key":"2026091606124096000_qkag042-B24","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1080\/01621459.2016.1148039","article-title":"A concave pairwise fusion approach to subgroup analysis","volume":"112","author":"Ma","year":"2017","journal-title":"Journal of the American Statistical Association"},{"issue":"2","key":"2026091606124096000_qkag042-B25","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1093\/biomet\/75.2.237","article-title":"Empirical likelihood ratio confidence intervals for a single functional","volume":"75","author":"Owen","year":"1988","journal-title":"Biometrika"},{"key":"2026091606124096000_qkag042-B26","volume-title":"Empirical likelihood","author":"Owen","year":"2001"},{"issue":"1","key":"2026091606124096000_qkag042-B27","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1214\/aos\/1176325370","article-title":"Empirical likelihood and general estimating equations","volume":"22","author":"Qin","year":"1994","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026091606124096000_qkag042-B28","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1093\/biomet\/asu048","article-title":"Using covariate-specific disease prevalence information to increase the power of case-control studies","volume":"102","author":"Qin","year":"2015","journal-title":"Biometrika"},{"key":"2026091606124096000_qkag042-B29","volume-title":"R: A language and environment for statistical computing","author":"R Core Team","year":"2023"},{"issue":"5","key":"2026091606124096000_qkag042-B30","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1137\/0314056","article-title":"Monotone operators and the proximal point algorithm","volume":"14","author":"Rockafellar","year":"1976","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"524","key":"2026091606124096000_qkag042-B31","doi-asserted-by":"publisher","first-page":"1698","DOI":"10.1080\/01621459.2017.1360779","article-title":"A massive data framework for M-estimators with cubic-rate","volume":"113","author":"Shi","year":"2018","journal-title":"Journal of the American Statistical Association"},{"issue":"7","key":"2026091606124096000_qkag042-B32","doi-asserted-by":"publisher","first-page":"1750","DOI":"10.1109\/TSP.2014.2304432","article-title":"On the linear convergence of the ADMM in decentralized consensus optimization","volume":"62","author":"Shi","year":"2014","journal-title":"IEEE Transactions on Signal Processing"},{"issue":"4","key":"2026091606124096000_qkag042-B33","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1093\/biomet\/asq057","article-title":"Penalized high-dimensional empirical likelihood","volume":"97","author":"Tang","year":"2010","journal-title":"Biometrika"},{"key":"2026091606124096000_qkag042-B34","author":"Tang","year":"2018"},{"issue":"1","key":"2026091606124096000_qkag042-B35","first-page":"3915","article-title":"Fused lasso approach in regression coefficients clustering\u2014Learning parameter heterogeneity in data integration","volume":"17","author":"Tang","year":"2016","journal-title":"Journal of Machine Learning Research: JMLR"},{"issue":"1","key":"2026091606124096000_qkag042-B36","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1214\/19-AOS1870","article-title":"Penalized generalized empirical likelihood with a diverging number of general estimating equations for censored data","volume":"48","author":"Tang","year":"2020","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026091606124096000_qkag042-B37","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression shrinkage and selection via the lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"issue":"1","key":"2026091606124096000_qkag042-B38","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1111\/j.1467-9868.2005.00490.x","article-title":"Sparsity and smoothness via the fused lasso","volume":"67","author":"Tibshirani","year":"2005","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2026091606124096000_qkag042-B39","first-page":"509","article-title":"Decentralized collaborative learning of personalized models over networks","author":"Vanhaesebrouck","year":"2017","journal-title":"Artificial intelligence and statistics"},{"issue":"3","key":"2026091606124096000_qkag042-B40","doi-asserted-by":"publisher","first-page":"1634","DOI":"10.1214\/18-AOS1730","article-title":"Distributed inference for quantile regression processes","volume":"47","author":"Volgushev","year":"2019","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026091606124096000_qkag042-B41","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1214\/07-AOS585","article-title":"Empirical likelihood for estimating equations with missing values","volume":"37","author":"Wang","year":"2009","journal-title":"Annals of Statistics"},{"issue":"2","key":"2026091606124096000_qkag042-B42","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1093\/biomet\/89.2.345","article-title":"Empirical likelihood-based inference in linear errors-in-covariables models with validation data","volume":"89","author":"Wang","year":"2002","journal-title":"Biometrika"},{"issue":"3","key":"2026091606124096000_qkag042-B43","doi-asserted-by":"publisher","first-page":"896","DOI":"10.1214\/aos\/1028674841","article-title":"Empirical likelihood-based inference under imputation for missing response data","volume":"30","author":"Wang","year":"2002","journal-title":"Annals of Statistics"},{"issue":"113","key":"2026091606124096000_qkag042-B44","first-page":"1","article-title":"Distributed inference for linear support vector machine","volume":"20","author":"Wang","year":"2019","journal-title":"Journal of Machine Learning Research"},{"issue":"3","key":"2026091606124096000_qkag042-B45","doi-asserted-by":"publisher","first-page":"806","DOI":"10.1080\/07350015.2002.2074426","article-title":"Network dradient descent algorithm for decentralized federated learning","volume":"41","author":"Wu","year":"2022","journal-title":"Journal of Business & Economic Statistics"},{"issue":"2","key":"2026091606124096000_qkag042-B46","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1214\/09-AOS729","article-title":"Nearly unbiased variable selection under minimax concave penalty","volume":"38","author":"Zhang","year":"2010","journal-title":"Annals of Statistics"},{"issue":"3","key":"2026091606124096000_qkag042-B47","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1093\/biomet\/asaa014","article-title":"Generalized integration model for improved statistical inference by leveraging external summary data","volume":"107","author":"Zhang","year":"2020","journal-title":"Biometrika"},{"issue":"4","key":"2026091606124096000_qkag042-B48","doi-asserted-by":"publisher","first-page":"2450015","DOI":"10.1142\/S2010326324500151","article-title":"Empirical likelihood inference for time-varying coefficient autoregressive models","volume":"13","author":"Zhang","year":"2024","journal-title":"Random Matrices: Theory and Applications"},{"issue":"545","key":"2026091606124096000_qkag042-B49","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1080\/01621459.2022.2126363","article-title":"Learning coefficient heterogeneity over networks: A distributed spanning-tree-based fused-lasso regression","volume":"119","author":"Zhang","year":"2022","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2026091606124096000_qkag042-B50","doi-asserted-by":"publisher","first-page":"2209","DOI":"10.5705\/ss.202020.0330","article-title":"Distributed empirical likelihood approach to integrating unbalanced datasets","volume":"33","author":"Zhou","year":"2023","journal-title":"Statistica Sinica"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/88\/4\/1303\/66841041\/qkag042.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/88\/4\/1303\/66841041\/qkag042.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T10:12:49Z","timestamp":1789553569000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/88\/4\/1303\/8471301"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,10]]},"references-count":50,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2026,2,10]]},"published-print":{"date-parts":[[2026,9,16]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkag042","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,9]]},"published":{"date-parts":[[2026,2,10]]}}}