{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:28:37Z","timestamp":1760146117949,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,5]],"date-time":"2024-10-05T00:00:00Z","timestamp":1728086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundations of China","award":["12271420","2024JC-YBMS-007","CXY-2021-117"],"award-info":[{"award-number":["12271420","2024JC-YBMS-007","CXY-2021-117"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["12271420","2024JC-YBMS-007","CXY-2021-117"],"award-info":[{"award-number":["12271420","2024JC-YBMS-007","CXY-2021-117"]}]},{"name":"Planning Project of Yulin Science and Technology Bureau of Shaanxi Province of China","award":["12271420","2024JC-YBMS-007","CXY-2021-117"],"award-info":[{"award-number":["12271420","2024JC-YBMS-007","CXY-2021-117"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Under the assumption of missing response data, empirical likelihood inference is studied via composite quantile regression. Firstly, three empirical likelihood ratios of composite quantile regression are given and proved to be asymptotically \u03c72. Secondly, without an estimation of the asymptotic covariance, confidence intervals are constructed for the regression coefficients. Thirdly, three estimators are presented for the regression parameters to obtain its asymptotic distribution. The finite sample performance is assessed through simulation studies, and the symmetry confidence intervals of the parametric are constructed. Finally, the effectiveness of the proposed methods is illustrated by analyzing a real-world data set.<\/jats:p>","DOI":"10.3390\/sym16101314","type":"journal-article","created":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T06:23:18Z","timestamp":1728282198000},"page":"1314","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Empirical Likelihood for Composite Quantile Regression Models with Missing Response Data"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1473-8458","authenticated-orcid":false,"given":"Shuanghua","family":"Luo","sequence":"first","affiliation":[{"name":"School of Science, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"},{"name":"Xi\u2019an International, Science and Technology Cooperation Base for Big Data Analysis and Algorithms, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Zheng","sequence":"additional","affiliation":[{"name":"School of Science, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6027-0606","authenticated-orcid":false,"given":"Cheng-yi","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Economics and Finance, Xi\u2019an Jiaotong University, Xi\u2019an 710061, China"},{"name":"System Behavior and Management Laboratory of Xi\u2019an Jiaotong University, Philosophy and Social Sciences Laboratory of the Ministry of Education in China, Xi\u2019an 710061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1007\/s00180-019-00886-y","article-title":"Weighted composite quantile regression for single index model with missing covariates at random","volume":"34","author":"Liu","year":"2019","journal-title":"Comput. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1080\/01621459.2014.928219","article-title":"Efficient quantile regression analysis with missing observations","volume":"110","author":"Chen","year":"2015","journal-title":"J. Am. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Luo, S.H., Yan, Y.X., and Zhang, C.Y. (2024). Two-Stage estimation of partially linear varying coefffcient quantile regression model with missing data. Mathematics, 12.","DOI":"10.3390\/math12040578"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107912","DOI":"10.1016\/j.csda.2023.107912","article-title":"Empirical likelihood in a partially linear single-index model with censored response data","volume":"193","author":"Xue","year":"2024","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Luo, S.H., Zhang, C.Y., and Wang, M.H. (2019). Composite quantile regression for varying coefficient models with response data missing at random. Symmetry, 11.","DOI":"10.3390\/sym11091065"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s10182-016-0278-8","article-title":"Smoothed empirical likelihood for quantile regression models with response data missing at random","volume":"15","author":"Luo","year":"2017","journal-title":"AStA-Adv. Stat. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1093\/biomet\/89.2.375","article-title":"Local multiple imputation","volume":"89","author":"Aerts","year":"2002","journal-title":"Biometrika"},{"key":"ref_8","unstructured":"Little, R.J.A., and Rubin, D.B. (2014). Statistical Analysis with Missing Data, John Wiley & Sons."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1037\/1082-989X.7.2.147","article-title":"Missing data: Our view of the state of the art","volume":"2","author":"Schafer","year":"2002","journal-title":"Psychol. Methods"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1080\/01621459.1995.10476493","article-title":"Analysis of semiparametric regression models for repeated outcomes in the presence of missing data","volume":"90","author":"Robins","year":"1995","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1198\/016214504000000449","article-title":"Semiparametric regression analysis with missing response at random","volume":"99","author":"Wang","year":"2004","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_12","first-page":"896","article-title":"Empirical likelihood-based inference under imputation for missing response data","volume":"30","author":"Wang","year":"2002","journal-title":"Ann. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1353","DOI":"10.1016\/j.jmva.2008.12.009","article-title":"Empirical likelihood for linear models with missing responses","volume":"100","author":"Xue","year":"2009","journal-title":"J. Multivar. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1214\/07-AOS507","article-title":"Composite quantile regression and the oracle model selection theory","volume":"36","author":"Zou","year":"2008","journal-title":"Ann. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s00362-014-0642-2","article-title":"Penalized weighted composite quantile estimators with missing covariates","volume":"57","author":"Yang","year":"2016","journal-title":"Stat. Pap."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s00180-020-01012-z","article-title":"Penalized weighted composite quantile regression for partially linear varying coefficient models with missing covariates","volume":"36","author":"Jin","year":"2020","journal-title":"Comput. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1080\/00949655.2022.2108030","article-title":"Composite quantile regression for heteroscedastic partially linear varying-coefficient models with missing censoring indicators","volume":"93","author":"Zou","year":"2023","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1214\/aos\/1176347494","article-title":"Empirical likelihood ratio confidence regions","volume":"18","author":"Owen","year":"1990","journal-title":"Ann. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1017\/S0266466606060087","article-title":"Smoothed empirical likelihood methods for quantile regression models","volume":"22","author":"Whang","year":"2006","journal-title":"Econom. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s12190-014-0804-3","article-title":"Empirical likelihood for composite quantile regression modeling","volume":"48","author":"Zhao","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2900","DOI":"10.1080\/03610926.2019.1678638","article-title":"Weighted composite quantile regression with censoring indicators missing at random","volume":"50","author":"Wang","year":"2021","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1080\/10485252.2016.1272692","article-title":"Empirical likelihood weighted composite quantile regression with partially missing covariates","volume":"29","author":"Sun","year":"2017","journal-title":"J. Nonparametric Stat."},{"key":"ref_23","first-page":"1","article-title":"Die productions and consumtionsver haltnisse des konigreichs sachsen","volume":"8","author":"Engel","year":"1857","journal-title":"Stat. Burdes"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1898","DOI":"10.1080\/03610920902923510","article-title":"Empirical likelihood inferences for semiparametric varying coefficient partially linear models with longitudinal data","volume":"39","author":"Zhao","year":"2010","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2933","DOI":"10.1016\/j.jspi.2009.01.016","article-title":"On locally weighted estimation and hypothesis testing of varying-coefficient models with missing covariates","volume":"139","author":"Wong","year":"2009","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1016\/j.jeconom.2007.08.016","article-title":"Conditional empirical likelihood estimation and inference for quantile regression models","volume":"142","author":"Otsu","year":"2008","journal-title":"J. Econom."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1314\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:11:03Z","timestamp":1760112663000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1314"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,5]]},"references-count":26,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["sym16101314"],"URL":"https:\/\/doi.org\/10.3390\/sym16101314","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,10,5]]}}}