{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:24:36Z","timestamp":1760232276168,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T00:00:00Z","timestamp":1666828800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundations of China","award":["11601409","2020JM571","2021JM-002"],"award-info":[{"award-number":["11601409","2020JM571","2021JM-002"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["11601409","2020JM571","2021JM-002"],"award-info":[{"award-number":["11601409","2020JM571","2021JM-002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The construction of confidence intervals is investigated for the partially linear varying coefficient quantile model with missing random responses. Combined with quantile regression, an imputation-based empirical likelihood method is proposed to construct confidence intervals for parametric and varying coefficient components. Then, it is proved that the proposed empirical log-likelihood ratios are asymptotically Chi-square in theory. Finally, the symmetry confidence intervals of the parametric components and the point-by-point confidence intervals of the varying coefficient components are constructed in the simulation studies to demonstrate further that the proposed method yields smaller confidence intervals and higher coverage probabilities.<\/jats:p>","DOI":"10.3390\/sym14112258","type":"journal-article","created":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T04:35:17Z","timestamp":1666845317000},"page":"2258","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Statistical Inference for Partially Linear Varying Coefficient Quantile Models with Missing Responses"],"prefix":"10.3390","volume":"14","author":[{"given":"Yuxin","family":"Yan","sequence":"first","affiliation":[{"name":"School of Science, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuanghua","family":"Luo","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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.3150\/bj\/1137421639","article-title":"Profile likelihood inferences on semiparametric varying coefficient partially linear models","volume":"11","author":"Fan","year":"2005","journal-title":"Bernoulli"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.spl.2004.01.018","article-title":"Wavelet estimation in varying coefficient partially linear regression models","volume":"68","author":"Zhou","year":"2004","journal-title":"Stat. 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