Strong consistency of estimators in partially linear models for longitudinal data with mixing-dependent structure

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English
18 pages
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For exhibiting dependence among the observations within the same subject, the paper considers the estimation problems of partially linear models for longitudinal data with the φ -mixing and ρ -mixing error structures, respectively. The strong consistency for least squares estimator of parametric component is studied. In addition, the strong consistency and uniform consistency for the estimator of nonparametric function are investigated under some mild conditions.

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Publié le 01 janvier 2011
Nombre de lectures 11
Langue English
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Zhou and Lin Journal of Inequalities and Applications 2011, 2011 :112 http://www.journalofinequalitiesandapplications.com/content/2011/1/112
R E S E A R C H Open Access Strong consistency of estimators in partially linear models for longitudinal data with mixing-dependent structure Xing-cai Zhou 1,2 and Jin-guan Lin 1*
* Correspondence: jglin@seu.edu.cn Abstract 1 Department of Mathematics, Southeast University, Nanjing For exhibiting dependence among the observations within the same subject, the 210096, People s Republic of China Full list of author information is paper considers the estimation problems of partially linear models for longitudinal available at the end of the article data with the -mixing and r -mixing error structures, respectively. The strong consistency for least squares estimator of parametric component is studied. In addition, the strong consistency and uniform consistency for the estimator of nonparametric function are investigated under some mild conditions. Keywords: partially linear model, longitudinal data, mixing dependent, strong consistency
1 Introduction Longitudinal data (Diggle et al. [1]) are cha racterized by repeated observations over time on the same set of individuals. They are common in medical and epidemiological studies. Examples of such data can be eas ily found in clinical trials and follow-up studies for monitoring disease progression . Interest of the study is often focused on evaluating the effects of time and covariates on the outcome variables. Let t ij be the time of the j th measurement of the i th subject, x ij Î R p and y ij be the i th subject s observed covariate and outcome at time t ij respectively. We assume that the full data-set { (x ij , y ij , t ij ), i = 1,..., n, j = 1,..., m i }, where n is the number of subjects and m i is the number of repeated measurements of the i th subject, is observed and can be modeled as the following partially linear models y ij = x iT β + g ( t ij ) + e ij (1 : 1) where b is a p × 1 vector of unknown parameter, g ( ) is an unknown smooth func-tion, e ij are random errors with E ( e ij ) = 0. We assume without loss of generality that t ij are all scaled into the interval I = [0, 1]. Although the observations, and therefore the e ij , from the different subjects are independent, they can be dependent within each subject. Partially linear models keep the flexibility of nonparametric models, while maintain-ing the explanatory power of parametric models (Fan and Li [2]). Many authors have studied the models in the form of (1.1) under some additional assumptions or restric-tions. If the nonparametric component g ( ) is known or not present in the models,
© 2011 Zhou and Lin; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.