{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T02:51:08Z","timestamp":1770346268146,"version":"3.49.0"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T00:00:00Z","timestamp":1746144000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T00:00:00Z","timestamp":1746144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Syst Sci Complex"],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s11424-025-5054-y","type":"journal-article","created":{"date-parts":[[2025,5,1]],"date-time":"2025-05-01T22:07:50Z","timestamp":1746137270000},"page":"309-333","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal Distributed Model Averaging for Multivariate Additive Model"],"prefix":"10.1007","volume":"39","author":[{"given":"Minghui","family":"Song","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianyao","family":"Qu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhihao","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guohua","family":"Zou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,2]]},"reference":[{"issue":"4","key":"5054_CR1","first-page":"1655","volume":"24","author":"X Chen","year":"2014","unstructured":"Chen X and Xie M G, A split-and-conquer approach for analysis of extraordinarily large data, Statistica Sinica, 2014, 24(4): 1655\u20131684.","journal-title":"Statistica Sinica"},{"key":"5054_CR2","first-page":"3299","volume":"16","author":"Y Zhang","year":"2015","unstructured":"Zhang Y, Duchi J, and Wainwright M, Divide and conquer kernel ridge regression: A distributed algorithm with minimax optimal rates, Journal of Machine Learning Research, 2015, 16: 3299\u20133340.","journal-title":"Journal of Machine Learning Research"},{"issue":"4","key":"5054_CR3","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1080\/10618600.2019.1586714","volume":"28","author":"G Xu","year":"2019","unstructured":"Xu G, Shang Z, and Cheng G, Distributed generalized cross-validation for divide-and-conquer kernel ridge regression and its asymptotic optimality, Journal of Computational and Graphical Statistics, 2019, 28(4): 891\u2013908.","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"113","key":"5054_CR4","first-page":"1","volume":"20","author":"X Wang","year":"2019","unstructured":"Wang X, Yang Z, Chen X, et al., Distributed inference for linear support vector machine, Journal of Machine Learning Research, 2019, 20(113): 1\u201341.","journal-title":"Journal of Machine Learning Research"},{"key":"5054_CR5","doi-asserted-by":"publisher","first-page":"104567","DOI":"10.1016\/j.jmva.2019.104567","volume":"176","author":"L Tang","year":"2020","unstructured":"Tang L, Zhou L, and Song P X K, Distributed simultaneous inference in generalized linear models via confidence distribution, Journal of Multivariate Analysis, 2020, 176: 104567.","journal-title":"Journal of Multivariate Analysis"},{"issue":"4","key":"5054_CR6","doi-asserted-by":"publisher","first-page":"1175","DOI":"10.1111\/j.1468-0262.2007.00785.x","volume":"75","author":"B E Hansen","year":"2007","unstructured":"Hansen B E, Least squares model averaging, Econometrica, 2007, 75(4): 1175\u20131189.","journal-title":"Econometrica"},{"issue":"2","key":"5054_CR7","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/j.jeconom.2009.10.030","volume":"156","author":"A T K Wan","year":"2010","unstructured":"Wan A T K, Zhang X, and Zou G, Least squares model averaging by Mallows criterion, Journal of Econometrics, 2010, 156(2): 277\u2013283.","journal-title":"Journal of Econometrics"},{"issue":"2","key":"5054_CR8","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.jeconom.2013.01.004","volume":"174","author":"X Zhang","year":"2013","unstructured":"Zhang X, Wan A T K, and Zou G, Model averaging by jackknife criterion in models with dependent data, Journal of Econometrics, 2013, 174(2): 82\u201394.","journal-title":"Journal of Econometrics"},{"issue":"1","key":"5054_CR9","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.jeconom.2014.11.005","volume":"188","author":"X Lu","year":"2015","unstructured":"Lu X and Su L, Jackknife model averaging for quantile regressions, Journal of Econometrics, 2015, 188(1): 40\u201358.","journal-title":"Journal of Econometrics"},{"issue":"2","key":"5054_CR10","first-page":"693","volume":"29","author":"X Zhang","year":"2019","unstructured":"Zhang X and Wang W, Optimal model averaging estimation for partially linear models, Statistica Sinica, 2019, 29(2): 693\u2013718.","journal-title":"Statistica Sinica"},{"issue":"1","key":"5054_CR11","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.jeconom.2018.10.007","volume":"209","author":"J Liao","year":"2019","unstructured":"Liao J, Zong X, Zhang X, et al., Model averaging based on leave-subject-out cross-validation for vector autoregressions, Journal of Econometrics, 2019, 209(1): 35\u201360.","journal-title":"Journal of Econometrics"},{"issue":"1","key":"5054_CR12","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1080\/07350015.2021.2006668","volume":"41","author":"R Zhu","year":"2022","unstructured":"Zhu R, Zhang X, Wan A T K, et al., Kernel averaging estimators, Journal of Business & Economic Statistics, 2022, 41(1): 157\u2013169.","journal-title":"Journal of Business & Economic Statistics"},{"issue":"1","key":"5054_CR13","first-page":"401","volume":"33","author":"J Liao","year":"2023","unstructured":"Liao J, Wan A T K, He S, et al., Frequentist model averaging for the nonparametric additive model, Statistica Sinica, 2023, 33(1): 401\u2013430.","journal-title":"Statistica Sinica"},{"issue":"215","key":"5054_CR14","first-page":"1","volume":"24","author":"H Zhang","year":"2023","unstructured":"Zhang H, Liu Z, and Zou G, Least squares model averaging for distributed data, Journal of Machine Learning Research, 2023, 24(215): 1\u201359.","journal-title":"Journal of Machine Learning Research"},{"issue":"2","key":"5054_CR15","doi-asserted-by":"publisher","first-page":"780","DOI":"10.1007\/s10114-025-3409-x","volume":"41","author":"H Zhang","year":"2025","unstructured":"Zhang H, Wan A T K, You K, et al., Distributed Mallows model averaging for ridge regressions, Acta Mathematica Sinica, English Series, 2025, 41(2): 780\u2013826.","journal-title":"Acta Mathematica Sinica, English Series"},{"issue":"161","key":"5054_CR16","first-page":"1","volume":"25","author":"X Liu","year":"2024","unstructured":"Liu X, Lian H, and Huang J, More efficient estimation of multivariate additive models based on tensor decomposition and penalization, Journal of Machine Learning Research, 2024, 25(161): 1\u201327.","journal-title":"Journal of Machine Learning Research"},{"issue":"1","key":"5054_CR17","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1080\/02331888.2017.1367794","volume":"52","author":"R Zhu","year":"2018","unstructured":"Zhu R, Zou G, and Zhang X, Model averaging for multivariate multiple regression models, Statistics, 2018, 52(1): 205\u2013227.","journal-title":"Statistics"},{"issue":"10","key":"5054_CR18","first-page":"3115","volume":"44","author":"T Qu","year":"2024","unstructured":"Qu T and Song M, Model averaging of multivariate linear regression models with heteroscedastic covariance matrix, Journal of Systems Science and Mathematical Sciences, 2024, 44(10): 3115\u20133132.","journal-title":"Journal of Systems Science and Mathematical Sciences"},{"issue":"2","key":"5054_CR19","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1214\/aos\/1176349548","volume":"13","author":"C J Stone","year":"1985","unstructured":"Stone C J, Additive regression and other nonparametric models, The Annals of Statistics, 1985, 13(2): 689\u2013705.","journal-title":"The Annals of Statistics"},{"key":"5054_CR20","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511618994","volume-title":"Spline Functions: Basic Theory","author":"L Schumaker","year":"2007","unstructured":"Schumaker L, Spline Functions: Basic Theory, Cambridge University Press, Cambridge, 2007."},{"issue":"5","key":"5054_CR21","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.1111\/j.1467-9868.2009.00718.x","volume":"71","author":"P Ravikumar","year":"2009","unstructured":"Ravikumar P, Lafferty J, Liu H, et al., Sparse additive models, Journal of the Royal Statistical Society Series B: Statistical Methodology, 2009, 71(5): 1009\u20131030.","journal-title":"Journal of the Royal Statistical Society Series B: Statistical Methodology"},{"key":"5054_CR22","doi-asserted-by":"publisher","first-page":"106902","DOI":"10.1016\/j.csda.2019.106902","volume":"144","author":"J Liao","year":"2020","unstructured":"Liao J and Zou G, Corrected Mallows criterion for model averaging, Computational Statistics and Data Analysis, 2020, 144: 106902.","journal-title":"Computational Statistics and Data Analysis"},{"key":"5054_CR23","first-page":"1281","volume":"19","author":"L Xue","year":"2009","unstructured":"Xue L, Consistent variable selection in additive models, Statistica Sinica, 2009, 19: 1281\u20131296.","journal-title":"Statistica Sinica"},{"issue":"2","key":"5054_CR24","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1111\/j.1369-7412.2004.05500.x","volume":"66","author":"J Z Huang","year":"2004","unstructured":"Huang J Z and Yang L, Identification of non-linear additive autoregressive models, Journal of the Royal Statistical Society Series B: Statistical Methodology, 2004, 66(2): 463\u2013477.","journal-title":"Journal of the Royal Statistical Society Series B: Statistical Methodology"},{"issue":"484","key":"5054_CR25","doi-asserted-by":"publisher","first-page":"1534","DOI":"10.1198\/016214508000000751","volume":"103","author":"H G M\u00fcller","year":"2008","unstructured":"M\u00fcller H G and Yao F, Functional additive models, Journal of the American Statistical Association, 2008, 103(484): 1534\u20131544.","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"5054_CR26","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1111\/rssc.12068","volume":"64","author":"S N Wood","year":"2015","unstructured":"Wood S N, Goude Y, and Shaw S, Generalized additive models for large data sets, Journal of the Royal Statistical Society Series C: Applied Statistics, 2015, 64(1): 139\u2013155.","journal-title":"Journal of the Royal Statistical Society Series C: Applied Statistics"},{"issue":"1","key":"5054_CR27","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1080\/10618600.2012.729985","volume":"23","author":"M W McLean","year":"2014","unstructured":"McLean M W, Hooker G, Staicu A M, et al., Functional generalized additive models, Journal of Computational and Graphical Statistics, 2014, 23(1): 249\u2013269.","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"4","key":"5054_CR28","first-page":"297","volume":"1","author":"T Hastie","year":"1986","unstructured":"Hastie T and Tibshirani R, Generalized additive models, Statistical Science, 1986, 1(4): 297\u2013310.","journal-title":"Statistical Science"},{"issue":"3","key":"5054_CR29","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1137\/1105028","volume":"5","author":"P Whittle","year":"1960","unstructured":"Whittle P, Bounds for the moments of linear and quadratic forms in independent variables, Theory of Probability and Its Applications, 1960, 5(3): 302\u2013305.","journal-title":"Theory of Probability and Its Applications"},{"issue":"3","key":"5054_CR30","doi-asserted-by":"publisher","first-page":"928","DOI":"10.1214\/009053604000000256","volume":"32","author":"J Fan","year":"2004","unstructured":"Fan J and Peng H, Nonconcave penalized likelihood with a diverging number of parameters, The Annals of Statistics, 2004, 32(3): 928\u2013961.","journal-title":"The Annals of Statistics"},{"issue":"522","key":"5054_CR31","doi-asserted-by":"publisher","first-page":"919","DOI":"10.1080\/01621459.2017.1302339","volume":"113","author":"J Chen","year":"2018","unstructured":"Chen J, Li D, Linton O, et al., Semiparametric ultra-high dimensional model averaging of nonlinear dynamic time series, Journal of the American Statistical Association, 2018, 113(522): 919\u2013932.","journal-title":"Journal of the American Statistical Association"}],"container-title":["Journal of Systems Science and Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-025-5054-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11424-025-5054-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-025-5054-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T12:45:26Z","timestamp":1770295526000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11424-025-5054-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,2]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["5054"],"URL":"https:\/\/doi.org\/10.1007\/s11424-025-5054-y","relation":{},"ISSN":["1009-6124","1559-7067"],"issn-type":[{"value":"1009-6124","type":"print"},{"value":"1559-7067","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,2]]},"assertion":[{"value":"17 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no conflict of interest.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}]}}