QIF Estimation In Partially Linear Models With Complex Data

The semiparametric models have become a class of important statistical models since1980s. This kind of models combines the advantages of parametric models with nonparametricmodels, and they have more implements and stronger explanations than pure parametric or non-parametric regression models. So a lot of actual problems can be described by these models,and these models will be better to fit the real data. In recent years, the studies on semiparamet-ric models become a hot research direction for many statisticians. With the rapid developmentof technology and computer sciences, the studies of the longitudinal data play more and moreimportant role in many fields, such as economy, medicine, finance, econometrics and so on. Inpractice, we often encounter some incomplete data, such as missing data, measurement errordata, censored data and so on. Hence, it has theoretical and practical significance to study thesemiparametric regression models through combining longitudinal data with incomplete data.Therefore, more and more statisticians have focused on the studies of the semiparametric re-gression models with complicated data.This paper mainly studies the the theory, methods and other related problems of the re-gression coefficients of semiparametric partially linear models under complex data by using thecorrected quadratic inference function method. The complex data we considered include twotypes. One type is the combination of the longitudinal data and measurement error data, andanother is the combination of the longitudinal data and missing data.The main works of this dissertation includes the following two aspects:1. We first focus on the efficient estimation for the marginal semiparametric partiallylinear models with longitudinal data when some covariates are measured with additive errors.We propose the corrected quadratic inference function method (QIF), and this method not onlyconsiders the within-cluster correlation but also corrects the bias from the measurement error.Under some regularity conditions, we show that the QIF estimator is consistent and asymptot-ically normal. Simulation studies are used to illustrate the proposed method under the limitedsamples. In addition, the proposed method is applied to the study of the real data collected formpatients infected with the virus of HIV in the years of1984to1991.2. For the marginal semiparametric partially linear models with longitudinal data, wepropose the corrected QIF method to study the efficient estimation of the parametric componentwhen the response variable is missing at random. The proposed method not only considersthe within-cluster correlation but also corrects the bias from the missing data. We estimate theregression coefficients based on complete data case and fractional imputation case, respectively.We conduct the simulation studies to deal with the missing data, and to illustrate the proposed two methods under the limited samples. We further compare the two simulated results.Finally, in conclusion and prospect, we summarize the main research achievements andinnovation acquired in this dissertation, and point out the further research questions and direc-tions.