{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T16:57:45Z","timestamp":1788973065597,"version":"build-2803163510"},"reference-count":38,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2006,5,15]],"date-time":"2006-05-15T00:00:00Z","timestamp":1147651200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Council of Hong Kong, Hong Kong, China","award":["HKBU7060\/04P"],"award-info":[{"award-number":["HKBU7060\/04P"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["10571008"],"award-info":[{"award-number":["10571008"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Beijing City of China","award":["1042002"],"award-info":[{"award-number":["1042002"]}]},{"name":"Science and Technology Development Project of the Education Committee of Beijing City","award":["KM200510005009"],"award-info":[{"award-number":["KM200510005009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2006,6,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Empirical-likelihood-based inference for the parameters in a partially linear single-index model is investigated. Unlike existing empirical likelihood procedures for other simpler models, if there is no bias correction the limit distribution of the empirical likelihood ratio cannot be asymptotically tractable. To attack this difficulty we propose a bias correction to achieve the standard \u03c72-limit. The bias-corrected empirical likelihood ratio shares some of the desired features of the existing least squares method: the estimation of the parameters is not needed; when estimating nonparametric functions in the model, undersmoothing for ensuring \u221an-consistency of the estimator of the parameters is avoided; the bias-corrected empirical likelihood is self-scale invariant and no plug-in estimator for the limiting variance is needed. Furthermore, since the index is of norm 1, we use this constraint as information to increase the accuracy of the confidence regions (smaller regions at the same nominal level). As a by-product, our approach of using bias correction may also shed light on nonparametric estimation in model checking for other semiparametric regression models. A simulation study is carried out to assess the performance of the bias-corrected empirical likelihood. An application to a real data set is illustrated.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2006.00556.x","type":"journal-article","created":{"date-parts":[[2006,5,15]],"date-time":"2006-05-15T11:27:27Z","timestamp":1147692447000},"page":"549-570","source":"Crossref","is-referenced-by-count":216,"title":["Empirical Likelihood Confidence Regions in a Partially Linear Single-Index Model"],"prefix":"10.1093","volume":"68","author":[{"given":"Lixing","family":"Zhu","sequence":"first","affiliation":[{"name":"Hong Kong Baptist University and Renmin University of China , Beijing , People's Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liugen","family":"Xue","sequence":"additional","affiliation":[{"name":"Beijing University of Technology , People's Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2006,5,15]]},"reference":[{"key":"2023040708492934200_","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1080\/01621459.1997.10474001","article-title":"Generalized partially linear single-index models","volume":"92","author":"Carroll","year":"1997","journal-title":"J. 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