{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,29]],"date-time":"2026-03-29T02:09:44Z","timestamp":1774750184467,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,8,21]],"date-time":"2019-08-21T00:00:00Z","timestamp":1566345600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the 217 Natural Science Foundations of Department of Shaanxi Province of China","award":["2017JK0344"],"award-info":[{"award-number":["2017JK0344"]}]},{"name":"the National Natural Science Foundations of China","award":["11601409"],"award-info":[{"award-number":["11601409"]}]},{"name":"the National Natural Science Foundations of China","award":["71501155"],"award-info":[{"award-number":["71501155"]}]},{"name":"the National Natural Science Foundations of China","award":["11201362"],"award-info":[{"award-number":["11201362"]}]},{"name":"the Natural Science Foundations of Shaanxi Province of China","award":["2016JM1009"],"award-info":[{"award-number":["2016JM1009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Composite quantile regression (CQR) estimation and inference are studied for varying coefficient models with response data missing at random. Three estimators including the weighted local linear CQR (WLLCQR) estimator, the nonparametric WLLCQR (NWLLCQR) estimator, and the imputed WLLCQR (IWLLCQR) estimator are proposed for unknown coefficient functions. Under some mild conditions, the proposed estimators are asymptotic normal. Simulation studies demonstrate that the unknown coefficient estimators with IWLLCQR are superior to the other two with WLLCQR and NWLLCQR. Moreover, bootstrap test procedures based on the IWLLCQR fittings is developed to test whether the coefficient functions are actually varying. Finally, a type of investigated real-life data is analyzed to illustrated the applications of the proposed method.<\/jats:p>","DOI":"10.3390\/sym11091065","type":"journal-article","created":{"date-parts":[[2019,8,21]],"date-time":"2019-08-21T11:19:06Z","timestamp":1566386346000},"page":"1065","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Composite Quantile Regression for Varying Coefficient Models with Response Data Missing at Random"],"prefix":"10.3390","volume":"11","author":[{"given":"Shuanghua","family":"Luo","sequence":"first","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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Meihua","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1111\/j.2517-6161.1993.tb01939.x","article-title":"Varying-coefficient models","volume":"55","author":"Hastie","year":"1993","journal-title":"J. R. Stat. Soc. Ser."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1198\/016214501753168280","article-title":"Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables","volume":"96","author":"Chiang","year":"2001","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1111\/j.1467-9868.2004.B5595.x","article-title":"Smoothing spline estimation in varying coefficient models","volume":"66","author":"Eubank","year":"2004","journal-title":"J. R. Stat. Soc. Ser."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1214\/aos\/1017939139","article-title":"Statistical estimation in varying coefficient models","volume":"27","author":"Fan","year":"1999","journal-title":"Ann. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1093\/biomet\/89.1.111","article-title":"Varying coefficient models and basis function approximations for the analysis of repeated measurements","volume":"89","author":"Huang","year":"2002","journal-title":"Biometrika"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1023\/A:1004125621021","article-title":"A two-step smoothing method for varying coefficient models with repeated measurements","volume":"52","author":"Wu","year":"2000","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.3150\/bj\/1137421639","article-title":"Profile likelihood inferences on semiparametric varying-cofficient partially linear models","volume":"11","author":"Fan","year":"2005","journal-title":"Bernoulli"},{"key":"ref_8","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_9","doi-asserted-by":"crossref","unstructured":"Koenker, R. (2005). Quantiles Regression, Cambridge University Press.","DOI":"10.1017\/CBO9780511754098"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1214\/009053606000000966","article-title":"Quantile regression with varying coefficients","volume":"35","author":"Kim","year":"2007","journal-title":"Ann. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1198\/016214508000000977","article-title":"Nonparametric quantile estimations for dynamic smooth coefficient models","volume":"103","author":"Cai","year":"2008","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/j.jeconom.2011.09.025","article-title":"Semiparametric quantile regression estimation in dynamic models with partially varying coefficients","volume":"167","author":"Cai","year":"2012","journal-title":"J. Econom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1007\/s00362-014-0629-z","article-title":"Robust estimation for spatial semiparametric varying coefficient partially linear regression","volume":"56","author":"Tang","year":"2015","journal-title":"Stat. Pap."},{"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":"305","DOI":"10.1214\/10-AOS842","article-title":"New efficient estimation and variable selection methods for semiparametric varying coefficient partially linear models","volume":"39","author":"Kai","year":"2011","journal-title":"Ann. Stat."},{"key":"ref_16","first-page":"1075","article-title":"New efficient and robust estimation in varying coefficient models with heteroscedasticity","volume":"22","author":"Guo","year":"2012","journal-title":"Stat. Sin."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1016\/j.jspi.2013.01.002","article-title":"Weighted local linear composite quantile estimation for the case of general error distributions","volume":"143","author":"Sun","year":"2013","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.jkss.2014.05.005","article-title":"Weighted composite quantile regression estimation and variable selection for varying coefficient models with heteroscedasticity","volume":"44","author":"Yang","year":"2015","journal-title":"J. Korean Stat. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s00362-015-0672-4","article-title":"Nonparametric M-type regression estimation under missing response data","volume":"57","author":"Luo","year":"2016","journal-title":"Stat. Pap."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1093\/biomet\/63.3.581","article-title":"Inference and missing data","volume":"63","author":"Rubin","year":"1976","journal-title":"Biometrika"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"b2393","DOI":"10.1136\/bmj.b2393","article-title":"Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls","volume":"338","author":"Sterne","year":"2009","journal-title":"BMJ"},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"1470","DOI":"10.1016\/j.jmva.2006.10.003","article-title":"Estimation in partially linear models with missing responses at random","volume":"98","author":"Wang","year":"2007","journal-title":"J. Multivar. Anal."},{"key":"ref_24","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_25","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1111\/j.1467-9469.2009.00651.x","article-title":"Empirical likelihood confidence intervals for response mean with data missing at random","volume":"36","author":"Xue","year":"2009","journal-title":"Scand. J. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1093\/biomet\/ass007","article-title":"Multiple imputation in quantile regression","volume":"99","author":"Wei","year":"2012","journal-title":"Biometrika"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s10182-013-0210-4","article-title":"Smoothed empirical likelihood analysis of partially linear quantile regression models with missing response variables","volume":"97","author":"Lv","year":"2013","journal-title":"Adv. Stat. Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4967","DOI":"10.1002\/sim.5883","article-title":"Weighted quantile regression for analyzing health care cost data with missing covariates","volume":"32","author":"Sherwood","year":"2013","journal-title":"Stat. Med."},{"key":"ref_29","first-page":"703","article-title":"Quantile regression for competing risks data with missing cause of failure","volume":"22","author":"Sun","year":"2012","journal-title":"Ann. Stat."},{"key":"ref_30","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. Assoc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"369","DOI":"10.4310\/SII.2013.v6.n3.a7","article-title":"Imputation methods for quantile estimation under missing at random","volume":"6","author":"Kim","year":"2013","journal-title":"Stat. Its Interface"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1117\/1.601136","article-title":"Nadaraya-Watson estimator for sensor fusion","volume":"36","author":"Nageswara","year":"1997","journal-title":"Opt. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2933","DOI":"10.1016\/j.jspi.2009.01.016","article-title":"On locally weighted estimation and hypothesis testing on varying coefficient models with missing covariates","volume":"139","author":"Wong","year":"2009","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1214\/aos\/1028144858","article-title":"Limiting distributions for L1 regression estimators under general conditions","volume":"26","author":"Knight","year":"1998","journal-title":"Ann. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1214\/aoms\/1177704472","article-title":"On estimation of a probability density function and model","volume":"33","author":"Parzen","year":"1962","journal-title":"Ann. Math. 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