Efcient Inference For The Varying Coefcient Mixed Efect Model With Longitudinal Data

Longitudinal data exists in many areas where measurements are taken withina subject repeatedly over time. Hence, data within the same subject is correlated.In the classical statistical inference, the sample is always assumed to be indepen-dent with each other. While in modelling the longitudinal data, the ignoranceof correlation within the same subject causes a loss of useful information and re-duces the efciency. The mixed efect model, in which random efect is includedto represent the individual correlation, is widely employed to analyze this kind ofcorrelated data. However, the linear mixed efect model under the assumption of alinear correlation between the response and covariate variables can’t character allkinds of complicated correlations. As a natural extension of the linear inference,the semiparametric mixed efect model has become increasingly popular in thelast decade. The semiparametric varying coefcient model not only possesses thesimpleness of the parametric model and the flexibility of the nonparametric model,but also can be implemented easily by various softwares. In the statistical infer-ence, it is a fundamental work to detect the important covariate variable. Whilemost existing variable selection methods have been confronted in the marginalmodel, little procedure is proposed to focus on the mixed efect model. Therefore,the research about the efcient estimation and variable selection for the varyingcoefcient mixed efect model is very theoretically significant and valuable. Morespecially, the research contents of this dissertation are summarized as follows:Combining the random efect with the B-spline smoothing technique, we con- sider an efcient estimation for the parametric and nonparametric component ofthe partially linear varying coefcient model with longitudinal data. By the as-sumption that the random efect and model error satisfy the variance componentmodel, we construct a consistent estimator for the covariance matrix, obtain an im-proved estimator for the parameter of interest. Given some regularized conditions,we prove that the estimators for the parametric, nonparametric and variance com-ponent are asymptotically normal and the asymptotic variance of the estimatorsfor the parametric component achieves the semiparametric bound. Some furthersimulations illustrate the validity of the proposed method and the improvementof the estimation efciency. Secondly, we propose an efcient shrinkage estima-tion method for simultaneous variable section and estimation when consideringthe within-subject correlation. With this newly proposed method, we can detectthe significant variables and estimate their coefcient simultaneously. Meanwhile,with a proper choice for the regularized parameters, we establish the consistencyand “Oracle” property of the estimators. Finally, several simulations assess theefect of the within-subject correlation on the variable selection and prove thesuperior sample performance.As to the varying coefcient mixed efect model, we firstly propose an unifiedefcient shrinkage procedure to select the parametric and nonparametric compo-nent with the consideration of the within-subject correlation. Therefore, we canselect the model and parameters simultaneously and reduce the risk of model mis-specification. Under some regularized assumption, we obtain the the consistencyand optimal convergence rate of the estimators and establish the asymptotic nor-mality for them. Simulation studies assess the finite sample performance in the end. Secondly, based on the generalized empirical likelihood method, we alsoconsider the construction of the confidence interval for the nonparametric com-ponent. Incorporating the within-subject correlation by the random efect, wepropose a generalized residual-adjusted block empirical likelihood ratio function.It is shown theoretically that the proposed likelihood ratio statistics is asymp-totically chi-squared without the undersmoothing condition. By this asymptoticresult, we further construct the confidence intervals for the nonparametric compo-nent and their simultaneous confidence regions for every two of them. Because ofthe consideration of the correlation, the robust property of the inference is beingsignificantly improved. Finally, some numerical simulations and an applicationillustrates that the proposed procedure performs well.As to the generalized partially linear model with random efect, we propose amethod that combines the efcient estimator with the random efect. Based on thevariance component model, we obtain a consistent estimator for the within-subjectcovariance matrix and develop a method to take account of the within-subject cor-relation with a proper way, by which we can make full use of the useful informationand obtain the efcient estimator without the fear of the extra bias. Moreover,when ensuring the convergence rate of the estimator for the covariance matrix, weshow that the error, incurred by the replacement of the estimator for the true co-variance matrix in the estimating equation, is asymptotic negligible. Meanwhile,with some given conditions, we prove that the estimator for the parametric com-ponent is asymptotic normal and semiparametric efcient. A simulation studyand application is used to reveal the improvement of the efciency.With the smoothing method of spline and local polynomial, we mainly consid- er the efcient estimation and variable selection with the semiparametric varyingcoefcient mixed model. Firstly, according to the model structure, we develop aproper method to incorporate the within-subject correlation and obtain a semi-parametric efcient estimator for the parameter of interest. Secondly, with theconsideration of random efect, we investigate the variable selection and empiricallikelihood inference for the varying coefcient model. Theoretical studies and nu-merical simulations show that the introduction of the within-subjection correlationhelp to improve the estimation efciency and enhance the inference robustness.The theory and methods in this dissertation enrich the method of efcient infer-ence for the varying coefcient model and its transformations, and expand greatlythe application fields of variable selection and empirical likelihood inference, whichis also help to simplify the model structure and improve the forecast accuracy.