Studies On Statistical Inference For Semiparametric Models With Complex Data

The analysis for nonparametric regression, if the dimension of covariate vari-ables is high, will cause the”curse of dimensionalty”. In order to solve this prob-lem, some semiparametric models are proposed and developed, such as single-indexmodels, partially linear single-index models, varying coefcient models. This pa-per is focused on the statistical inference for these semiparametric models withcomplex data. The research mainly follows several aspects:Firstly, based on the generalized estimating equations (GEE) and thequadratic inference functions (QIF) methods, we propose a bias-corrected gener-alized empirical likelihood to make statistical inference for the single-index modelwith longitudinal data. The maximum empirical likelihood estimator and the bias-corrected generalized empirical log-likelihood ratio statistics for the unknown indexparameter in the model are obtained. It is proved that the maximum empiricallikelihood estimator is asymptotic normal and the proposed statistics are asymp-totically chi-square distribution under some suitable conditions, and hence theycan be used to construct the confdence region of the index parameter. Throughsimulation studies, we compare the empirical likelihood confdence region by theproposed method with the confdence region obtained by other methods.Secondly, based on the GEE and the local polynomial regression method, weconstruct the simultaneous confdence band of the link function in single-indexrandom efects models with longitudinal data, and obtain the estimator of thelink function. We further establish the asymptotic properties of the link function and its derivative under some regularity conditions and√n-consistent estimatorsof the index parameter vector. The results can be used to build the simultaneousconfdence band of the link function for various inference tasks. We also conductsimulation studies to evaluate the performance of the proposed method, and applythem to a real data set for illustration.Thirdly, based on the smooth-threshold GEE and the local polynomial re-gression method, we consider the problem of variable selection for the single-indexrandom efects models with longitudinal data. The proposed method shares someof the desired features of existing variable selection methods: the resulting es-timator enjoys the oracle property; the proposed procedure avoids the convexoptimization problem and is fexible and easy to implement. Moreover, we usethe penalized weighted deviance criterion for a data-driven choice of the tuningparameters. Simulation studies are carried out to assess the performance of ourmethod, and a real dataset is analyzed for further illustrationFourthly, based on qunatile regression method and local quantile regressionmethod, we develop estimation procedures for partially linear single-index qunatileregression models, obtain the estimator of unknown link function and the esti-mator of index parameters and the parameters in the linear component of themodel. Under some regularity conditions, we show that the proposed estimatorsare asymptotically normal. A simulation study is conducted to evaluate the per-formance of the proposed method and a real data set is analyzed as an illustration.Lastly, for Tecator data, we introduce a partial functional linear varying co-efcient model. The spline estimators of the unknown functional coefcient are obtained, and the rates of convergence of the spline estimators and the meansquared prediction error are proved under some appropriate conditions. A sim-ulation study is conducted to illustrate the performance of our method. Finally,we analyze the Tecator data by using the proposed model in this paper. With thecomparison with other models, we can fnd that the model proposed in this paperis superior than other models.