{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T17:13:51Z","timestamp":1778087631897,"version":"3.51.4"},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T00:00:00Z","timestamp":1617580800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T00:00:00Z","timestamp":1617580800000},"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":[[2022,2]]},"DOI":"10.1007\/s11424-021-0158-5","type":"journal-article","created":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T05:02:46Z","timestamp":1617598966000},"page":"264-282","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Model Averaging Estimation for Varying-Coefficient Single-Index Models"],"prefix":"10.1007","volume":"35","author":[{"given":"Yue","family":"Liu","sequence":"first","affiliation":[]},{"given":"Jiahui","family":"Zou","sequence":"additional","affiliation":[]},{"given":"Shangwei","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Qinglong","family":"Yang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,5]]},"reference":[{"issue":"448","key":"158_CR1","doi-asserted-by":"publisher","first-page":"1275","DOI":"10.1080\/01621459.1999.10473880","volume":"94","author":"Y Xia","year":"1999","unstructured":"Xia Y and Li W K, On single-index coefficient regression models, Journal of the American Statistical Association, 1999, 94(448): 1275\u20131285.","journal-title":"Journal of the American Statistical Association"},{"key":"158_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-17146-8","volume-title":"Nonparametric and Semiparametric Models","author":"W H\u00e4rdle","year":"2004","unstructured":"H\u00e4rdle W, M\u00fcller M, Sperlich S, et al., Nonparametric and Semiparametric Models, Springer-Verlag, Berlin, 2004."},{"issue":"3","key":"158_CR3","doi-asserted-by":"publisher","first-page":"1458","DOI":"10.1016\/j.csda.2007.04.008","volume":"52","author":"H Wong","year":"2008","unstructured":"Wong H, Wai C I, and Zhang R, Varying-coefficient single-index model, Computational Statistics & Data Analysis, 2008, 52(3): 1458\u20131476.","journal-title":"Computational Statistics & Data Analysis"},{"issue":"9","key":"158_CR4","doi-asserted-by":"publisher","first-page":"4311","DOI":"10.1080\/03610926.2015.1081950","volume":"46","author":"Y Zhao","year":"2016","unstructured":"Zhao Y, X\u00fce L, and Feng S, Semiparametric estimation of the single-index varying-coefficient model, Communications in Statistics \u2014 Theory and Methods, 2016, 46(9): 4311\u20134326.","journal-title":"Communications in Statistics \u2014 Theory and Methods"},{"issue":"1","key":"158_CR5","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/0304-4076(93)90114-K","volume":"58","author":"H Ichimura","year":"1993","unstructured":"Ichimura H, Semiparametric least squares and weighted sls estimation of single-index models, Journal of Econometrics, 1993, 58(1): 71\u2013120.","journal-title":"Journal of Econometrics"},{"issue":"1","key":"158_CR6","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1214\/aos\/1176349020","volume":"21","author":"W H\u00e4rdle","year":"1993","unstructured":"H\u00e4rdle W, Hall P, and Ichimura H, Optimal smoothing in single-index models, The Annals of Statistics, 1993, 21(1): 157\u2013178.","journal-title":"The Annals of Statistics"},{"issue":"6","key":"158_CR7","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.2307\/1914309","volume":"54","author":"T M Stoker","year":"1986","unstructured":"Stoker T M, Consistent estimation of scaled coefficients, Econometrica, 1986, 54(6): 1461\u20131481.","journal-title":"Econometrica"},{"issue":"408","key":"158_CR8","first-page":"986","volume":"84","author":"W H\u00e4rdle","year":"1989","unstructured":"H\u00e4rdle W and Stoker T M, Investigating smooth multiple regression by the method of average derivatives, Journal of the American Statistical Association, 1989, 84(408): 986\u2013995.","journal-title":"Journal of the American Statistical Association"},{"issue":"414","key":"158_CR9","doi-asserted-by":"publisher","first-page":"316","DOI":"10.1080\/01621459.1991.10475035","volume":"86","author":"K C Li","year":"1991","unstructured":"Li K C, Sliced inverse regression for dimension reduction, Journal of the American Statistical Association, 1991, 86(414): 316\u2013327.","journal-title":"Journal of the American Statistical Association"},{"key":"158_CR10","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/BF01437407","volume":"24","author":"G Wahba","year":"1975","unstructured":"Wahba G, Smoothing noisy data with spline function, Numerische Mathematik, 1975, 24: 383\u2013393.","journal-title":"Numerische Mathematik"},{"issue":"1","key":"158_CR11","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1111\/j.2517-6161.1983.tb01239.x","volume":"45","author":"G Wahba","year":"1983","unstructured":"Wahba G, Bayesian confidence-intervals for the cross-validated smoothing spline, Journal of the Royal Statistical Society, Series B, 1983, 45(1): 133\u2013150.","journal-title":"Journal of the Royal Statistical Society, Series B"},{"issue":"4","key":"158_CR12","first-page":"1378","volume":"13","author":"G Wahba","year":"1985","unstructured":"Wahba G, A comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem, The Annals of Mathematical Statistics, 1985, 13(4): 1378\u20131402.","journal-title":"The Annals of Mathematical Statistics"},{"issue":"1","key":"158_CR13","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1137\/1109020","volume":"9","author":"E A Nadaraya","year":"1964","unstructured":"Nadaraya E A, On estimating regression, Theory Probability and Its Application, 1964, 9(1): 141\u2013142.","journal-title":"Theory Probability and Its Application"},{"issue":"4","key":"158_CR14","first-page":"359","volume":"26","author":"G S Watson","year":"1964","unstructured":"Watson G S, Smooth regression analysis, Sankhya: The Indian Journal of Statistics, Series A, 1964, 26(4): 359\u2013372.","journal-title":"Sankhya: The Indian Journal of Statistics, Series A"},{"issue":"420","key":"158_CR15","doi-asserted-by":"publisher","first-page":"998","DOI":"10.1080\/01621459.1992.10476255","volume":"87","author":"J Fan","year":"1991","unstructured":"Fan J, Design adaptive nonparametric regression, Journal of the American Statistical Association, 1991, 87(420): 998\u20131004.","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"158_CR16","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1214\/aos\/1176349022","volume":"21","author":"J Fan","year":"1993","unstructured":"Fan J, Local linear regression smoothers and their minimax efficiency, The Annals of Statistics, 1993, 21(1): 196\u2013216.","journal-title":"The Annals of Statistics"},{"key":"158_CR17","volume-title":"Local Polynomial Modeling and Its Application","author":"J Fan","year":"1996","unstructured":"Fan J and Gijbels I, Local Polynomial Modeling and Its Application, Chapman and Hall, London, 1996."},{"issue":"460","key":"158_CR18","doi-asserted-by":"publisher","first-page":"1042","DOI":"10.1198\/016214502388618861","volume":"97","author":"Y Yu","year":"2002","unstructured":"Yu Y and Ruppert D, Penalized spline estimation for partially linear single-index models, Journal of the American Statistical Association, 2002, 97(460): 1042\u20131054.","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"158_CR19","doi-asserted-by":"publisher","first-page":"6","DOI":"10.1007\/s11460-011-0126-2","volume":"6","author":"Z H Zhou","year":"2011","unstructured":"Zhou Z H, When semi-supervised learning meets ensemble learning, Frontiers of Electrical & Electronic Engineering in China, 2011, 6(1): 6\u201316.","journal-title":"Frontiers of Electrical & Electronic Engineering in China"},{"key":"158_CR20","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.dss.2014.10.004","volume":"68","author":"E Fersini","year":"2014","unstructured":"Fersini E, Messina E, and Pozzi F A, Sentiment analysis: Bayesian ensemble learning, Decision Support Systems, 2014, 68: 26\u201338.","journal-title":"Decision Support Systems"},{"issue":"1","key":"158_CR21","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1093\/bioinformatics\/btw539","volume":"33","author":"B Liu","year":"2017","unstructured":"Liu B, Wang S, Ren L, et al., irspot-el: Identify recombination spots with an ensemble learning approach, Bioinformatics, 2017, 33(1): 35\u201341.","journal-title":"Bioinformatics"},{"issue":"4","key":"158_CR22","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":"516","key":"158_CR23","doi-asserted-by":"publisher","first-page":"1775","DOI":"10.1080\/01621459.2015.1115762","volume":"111","author":"X Zhang","year":"2016","unstructured":"Zhang X, Yu D, Zou G, et al., Optimal model averaging estimation for generalized linear models and generalized linear mixed-effects models, Journal of the American Statistical Association, 2016, 111(516): 1775\u20131790.","journal-title":"Journal of the American Statistical Association"},{"key":"158_CR24","first-page":"610","volume":"1","author":"H Akaike","year":"1973","unstructured":"Akaike H, Information theory and an extension of the maximum likelihood principle, Breakthroughs in Statistics, 1973, 1: 610\u2013624.","journal-title":"Breakthroughs in Statistics"},{"issue":"2","key":"158_CR25","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1214\/aos\/1176344136","volume":"6","author":"G Schwarz","year":"1978","unstructured":"Schwarz G, Estimating the dimension of a model, The Annals of Statistics, 1978, 6(2): 461\u2013464.","journal-title":"The Annals of Statistics"},{"issue":"1","key":"158_CR26","first-page":"87","volume":"42","author":"C L Mallows","year":"1973","unstructured":"Mallows C L, Some comments on cp, Technometrics, 1973, 42(1): 87\u201394.","journal-title":"Technometrics"},{"issue":"2","key":"158_CR27","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":"526","key":"158_CR28","doi-asserted-by":"publisher","first-page":"882","DOI":"10.1080\/01621459.2018.1456936","volume":"114","author":"R Zhu","year":"2019","unstructured":"Zhu R, Wan A T K, Zhang X, et al., A mallows-type model averaging estimator for the varying-coefficient partially linear model, Journal of the American Statistical Association, 2019, 114(526): 882\u2013892.","journal-title":"Journal of the American Statistical Association"},{"key":"158_CR29","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1214\/ss\/1009212519","volume":"14","author":"J A Hoeting","year":"1999","unstructured":"Hoeting J A, Madigan D, Raftery A E, et al., Bayesian model averaging: A tutorial, Statistical Science, 1999, 14: 382\u2013417.","journal-title":"Statistical Science"},{"issue":"4","key":"158_CR30","doi-asserted-by":"publisher","first-page":"732","DOI":"10.1007\/s11424-009-9198-y","volume":"22","author":"H Wang","year":"2009","unstructured":"Wang H, Zhang X, and Zou G, Frequentist model averaging estimation: A review, Journal of Systems Science and Complexity, 2009, 22(4): 732\u2013748.","journal-title":"Journal of Systems Science and Complexity"},{"key":"158_CR31","doi-asserted-by":"publisher","first-page":"603","DOI":"10.2307\/2533961","volume":"53","author":"S T Buckland","year":"1997","unstructured":"Buckland S T, Burnham K P, and Augustin N H, Model selection: An integral part of inference, Biometrics, 1997, 53: 603\u2013618.","journal-title":"Biometrics"},{"key":"158_CR32","doi-asserted-by":"publisher","first-page":"972","DOI":"10.1111\/j.1541-0420.2006.00567.x","volume":"62","author":"G Claeskens","year":"2006","unstructured":"Claeskens G, Croux C, and van K J, Variable selection for logistic regression using a prediction-focused information criterion, Biometrics, 2006, 62: 972\u2013979.","journal-title":"Biometrics"},{"key":"158_CR33","doi-asserted-by":"publisher","first-page":"1053","DOI":"10.1198\/jasa.2011.tm09478","volume":"106","author":"H Liang","year":"2011","unstructured":"Liang H, Zou G, Wan A T K, et al., Optimal weight choice for frequentist model average estimators, Journal of the American Statistical Association, 2011, 106: 1053\u20131066.","journal-title":"Journal of the American Statistical Association"},{"issue":"6","key":"158_CR34","doi-asserted-by":"publisher","first-page":"2013","DOI":"10.1007\/s11424-020-9343-1","volume":"33","author":"Z H Zhao","year":"2020","unstructured":"Zhao Z H and Zou G H, Average estimation of semiparametric models for high-dimensional longitudinal data, Journal of Systems Science and Complexity, 2020, 33(6): 2013\u20132047.","journal-title":"Journal of Systems Science and Complexity"},{"issue":"4","key":"158_CR35","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s11424-018-7093-0","volume":"32","author":"X Zhang","year":"2019","unstructured":"Zhang X, Zheng Y, and Wang S, A demand forecasting method based on stochastic frontier analysis and model average: An application in air travel demand forecasting, Journal of Systems Science and Complexity, 2019, 32(4): 615\u2013633.","journal-title":"Journal of Systems Science and Complexity"},{"key":"158_CR36","doi-asserted-by":"publisher","first-page":"1202","DOI":"10.1198\/016214505000000088","volume":"100","author":"Z Yuan","year":"2005","unstructured":"Yuan Z and Yang Y, Combining linear regression models: When and how. Journal of the American Statistical Association, 2005, 100: 1202\u20131214.","journal-title":"Journal of the American Statistical Association"},{"key":"158_CR37","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.jeconom.2013.04.019","volume":"176","author":"X Zhang","year":"2013","unstructured":"Zhang X, Lu Z, and Zou G, Adaptively combined forecasting for discrete response time series, Journal of Econometrics, 2013, 176: 80\u201391.","journal-title":"Journal of Econometrics"},{"key":"158_CR38","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.jeconom.2014.07.002","volume":"186","author":"C Liu","year":"2015","unstructured":"Liu C, Distribution theory of the least squares averaging estimator, Journal of Econometrics, 2015, 186: 142\u2013159.","journal-title":"Journal of Econometrics"},{"key":"158_CR39","doi-asserted-by":"crossref","unstructured":"Yin S, Liu C, and Lin C, Focused information criterion and model averaging for large panels with a multifactor error structure, Journal of Business & Economic Statistics, 2019, forthcoming.","DOI":"10.1080\/07350015.2019.1623044"},{"key":"158_CR40","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1016\/j.jeconom.2015.02.025","volume":"187","author":"D Li","year":"2015","unstructured":"Li D, Linton O, and Lu Z, A flexible semiparametric forecasting model for time series, Journal of Econometrics, 2015, 187: 345\u2013357.","journal-title":"Journal of Econometrics"},{"issue":"3","key":"158_CR41","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1214\/aos\/1009210682","volume":"29","author":"M Hristache","year":"2001","unstructured":"Hristache M, Juditsky A, and Spokoiny V, Direct estimation of the index coefficient in a single-index model, The Annals of Statistics, 2001, 29(3): 595\u2013623.","journal-title":"The Annals of Statistics"},{"issue":"5","key":"158_CR42","doi-asserted-by":"publisher","first-page":"1491","DOI":"10.1214\/aos\/1017939139","volume":"27","author":"J Fan","year":"1999","unstructured":"Fan J and Zhang W, Statistical estimation in varying coefficient models, The Annals of Statistics, 1999, 27(5): 1491\u20131518.","journal-title":"The Annals of Statistics"},{"issue":"1","key":"158_CR43","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.jeconom.2011.06.019","volume":"167","author":"B E Hansen","year":"2012","unstructured":"Hansen B E and Racine J S, Jackknife model averaging, Journal of Econometrics, 2012, 167(1): 38\u201346.","journal-title":"Journal of Econometrics"},{"issue":"2","key":"158_CR44","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":"464","key":"158_CR45","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1198\/016214503000000828","volume":"98","author":"N L Hjort","year":"2003","unstructured":"Hjort N L and Claeskens G, Frequentist model average estimators, Journal of the American Statistical Association, 2003, 98(464): 879\u2013899.","journal-title":"Journal of the American Statistical Association"},{"issue":"464","key":"158_CR46","doi-asserted-by":"publisher","first-page":"938","DOI":"10.1198\/016214503000000882","volume":"98","author":"N L Hjort","year":"2003","unstructured":"Hjort N L and Claeskens G, Rejoinder to the discussion of \u201cfrequentist model average estimators\u201d and \u201cfocused information criterion\u201d, Journal of the American Statistical Association, 2003, 98(464): 938\u2013945.","journal-title":"Journal of the American Statistical Association"},{"issue":"476","key":"158_CR47","doi-asserted-by":"publisher","first-page":"1449","DOI":"10.1198\/016214506000000069","volume":"101","author":"N L Hjort","year":"2006","unstructured":"Hjort N L and Claeskens G, Focused information criteria and model averaging for the cox hazard regression model, Journal of the American Statistical Association, 2006, 101(476): 1449\u20131464.","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"158_CR48","doi-asserted-by":"publisher","first-page":"958","DOI":"10.1214\/aos\/1176350486","volume":"15","author":"K C Li","year":"1987","unstructured":"Li K C, Asymptotic optimality for cp, cl, cross-validation and generalized cross-validation: Discrete dindex set, The Annals of Statistics, 1987, 15(3): 958\u2013975.","journal-title":"The Annals of Statistics"},{"issue":"4","key":"158_CR49","first-page":"2795","volume":"28","author":"C Li","year":"2018","unstructured":"Li C, Li Q, Racine J S, et al., Optimal model averaging of varying coefficient models, Statistica Sinica, 2018, 28(4): 2795\u20132809.","journal-title":"Statistica Sinica"},{"key":"158_CR50","doi-asserted-by":"publisher","first-page":"4156","DOI":"10.1080\/03610926.2012.719989","volume":"43","author":"Z Yu","year":"2014","unstructured":"Yu Z, He B, and Chen M, Empirical likelihood for generalized partially linear single-index models, Communications in Statistics, 2014, 43: 4156\u20134163.","journal-title":"Communications in Statistics"},{"key":"158_CR51","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1016\/j.csda.2015.02.011","volume":"88","author":"S Xie","year":"2015","unstructured":"Xie S, Wan A T K, and Zhou Y, Quantile regression methods with varying-coefficient models for censored data, Computational Statistics & Data Analysis, 2015, 88: 154\u2013172.","journal-title":"Computational Statistics & Data Analysis"},{"issue":"2","key":"158_CR52","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s10463-017-0642-9","volume":"71","author":"Y Gao","year":"2018","unstructured":"Gao Y, Zhang X, Wang S, et al., Frequentist model averaging for threshold models, Annals of the Institute of Statistical Mathematics, 2018, 71(2): 275\u2013306.","journal-title":"Annals of the Institute of Statistical Mathematics"},{"key":"158_CR53","unstructured":"Zhang X, Model averaging and its applications, PhD thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 2010."}],"container-title":["Journal of Systems Science and Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-021-0158-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11424-021-0158-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-021-0158-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T14:47:40Z","timestamp":1724770060000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11424-021-0158-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,5]]},"references-count":53,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["158"],"URL":"https:\/\/doi.org\/10.1007\/s11424-021-0158-5","relation":{},"ISSN":["1009-6124","1559-7067"],"issn-type":[{"value":"1009-6124","type":"print"},{"value":"1559-7067","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,5]]},"assertion":[{"value":"15 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 October 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 April 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}