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The derivative is used to show the asymptotic normality of the estimator with the SE expressed in terms of the profile likelihood score function.<\/jats:p>","DOI":"10.3390\/e22030278","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T04:16:16Z","timestamp":1583122576000},"page":"278","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Statistical Generalized Derivative Applied to the Profile Likelihood Estimation in a Mixture of Semiparametric Models"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5565-3314","authenticated-orcid":false,"given":"Yuichi","family":"Hirose","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Victoria University of Wellington, P.O. 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Joint Models for Longitudinal and Time-to-Event Data With Applications in R, CRC Press.","DOI":"10.1201\/b12208"},{"key":"ref_3","first-page":"809","article-title":"Joint modeling of longitudinal and time to event data: An overview","volume":"14","author":"Tsiatis","year":"2004","journal-title":"Stat. Sin."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"330","DOI":"10.2307\/2533118","article-title":"A joint model for survival and longitudinal data measured with error","volume":"53","author":"Wulfsohn","year":"1997","journal-title":"Biometrics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1093\/biostatistics\/3.1.33","article-title":"Identification and efficiency of longitudinal markers for survival","volume":"3","author":"Henderson","year":"2002","journal-title":"Biostatistics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1111\/j.0006-341X.2004.00244.x","article-title":"Joint modeling of longitudinal and survival data via a common frailty","volume":"60","author":"Ratcliffe","year":"2004","journal-title":"Biometrics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1111\/j.0006-341X.2002.00742.x","article-title":"A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data","volume":"58","author":"Song","year":"2002","journal-title":"Biometrika"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2132","DOI":"10.1214\/009053605000000480","article-title":"Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time","volume":"33","author":"Zeng","year":"2005","journal-title":"Ann. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1111\/j.1369-7412.2007.00606.x","article-title":"Maximum likelihood estimation in semiparametric regression models with censored data","volume":"69","author":"Zeng","year":"2007","journal-title":"J. R. Stat. Soc.: Series B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1198\/016214506000001239","article-title":"Semiparametric Transformation Models With Random Effects for Recurrent Events","volume":"102","author":"Zeng","year":"2007","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_11","first-page":"871","article-title":"A general asymptotic theory for maximum likelihood estimation in semiparametric regression models with censored data","volume":"20","author":"Zeng","year":"2010","journal-title":"Stat. Sin."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1080\/01621459.2000.10474219","article-title":"On profile likelihood (with discussion)","volume":"95","author":"Murphy","year":"2000","journal-title":"J. Amer. Statist. Assoc."},{"key":"ref_13","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"van der Vaart, A.W. (1998). Asymptotic Statistics, Cambridge University Press.","DOI":"10.1017\/CBO9780511802256"},{"key":"ref_15","unstructured":"Kolmogorov, A.N., and Fomin, S.V.e. (1975). Introductory Real Analysis, Dover Publication."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"van der Vaart, A.W., and Wellner, J.A. (1996). Weak Convergence and Empirical Processes, Springer.","DOI":"10.1007\/978-1-4757-2545-2"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1111\/anzs.12153","article-title":"Joint modeling of survival and longitudinal ordered data using a semiparametric approach","volume":"58","author":"Preedalikit","year":"2016","journal-title":"Aust. New Zealand J. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Kalbfleisch, J.D., and Prentice, R.L. (2002). The Statistical Analysis of Failure Time Data, John Wiley & Sons. 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