Statistical Analysis For Semiparametric Varying-coe±cient Partially Linear Models With Incomplete Data

Semiparametric varying-coe?cient partially linear (SVCPL) models exten-sively cover some important semiparametric models such as partially linear mod-els, varying-coe?cient models and so on. Their advantages lie in their goodcombination of two-folds. On one hand, they have the merits of linear modelwhich are prone to easily interpretation, constructing estimation and tests. Onthe other hand, they display robust and ?exible virtues as for nonparametricmodels. Furthermore, their varying-coe?cient part may expound an interactionbetween covariates, dynamic changes (e.g. when varying-coe?cients are relatedto time), reduction of the dimension of data and so on. Therefore they are widelyapplied in economics, finance, biomedicine and some other fields.However, in statistical analysis, we often have to deal with incomplete datasuch as missing data and measurement errors due to the high cost of experiments,the imprecise measurement of equipments, and refusal of replying from investi-gators or other reasons. This would arise many problems if the usual statisticalmodels and methods are directly applied to those data, such as having a largebias in statistical estimation and sometimes, even leading to completely-reversedtesting results. On these grounds, the thesis will focus on studies about the statis-tical inference of semiparametric varying-coe?cient partially linear models whenincomplete data exist in variables and systematically discuss the estimation pro-cedure of these models for their parametric and nonparametric varying-coe?cientpart in three kinds of incomplete data so that a new perspective may be viewedfor their relevant theory and practice.The thesis is divided into four chapters. Chapter One contains introductionand relevant literature review. In Chapter Two, the statistical inference problemof SVCPL models is dwelled upon when their covariates in nonparametric part aremissing. More specifically, this covariate is precisely measured on validation setbut missing in primary data set whereas there are auxiliary information providedby a surrogate variable for this covariate. The method is put forward to cor- rect (or update) estimators of semiparametric profile least-squared methods forparametric and nonparametric varying-coe?cient components in order to incor-porate auxiliary information to the validation data set. The resulting estimatorsare consistent and show asymptotic normality regardless of the specification ofthe relationship between true covariates and the surrogate variables. They arealso more asymptotically e?cient than the validation-set-only estimators.In Chapter Three, the statistical analysis of SVCPL models focuses on thecase when their response variable is missing and some of their covariates haveadditive measurement errors. Using the completely observed samples, the esti-mators of parametric and nonparametric varying-coe?cient components are ob-tained via semiparametric profile least-squared methods. Then, the estimatorsare modified by correcting the attenuation which comes from the additive mea-surement errors in covariates. Asymptotic properties of the corrected estimatorsare established. Simulations of the finite sample performance are investigatedand the results show that our estimated methods are reasonable and havinggood properties. In the end, the procedure for estimating the missing probabilityof response variable is proposed via the usage of the additive measurement errormodel for covariates.Then, based on the inverse probability weighted estimat-ing equations, a method to improve estimators of parametric and nonparametriccomponents obtained before is suggested.And Chapter Four mainly discusses the estimation of SVCPL models whenthere appear Berkson measurement errors in their covariates. The major di?er-ence between Berkson measurement errors and additive measurement errors isthat for linear models, Berkson measurement errors will not cause a bias whenthe parameters are estimated. But for nonlinear and nonparametric models,Berkson measurement errors may lead to large biases for estimators. However,the research in this part shows that for SVCPL models, even if there are Berksonmeasurement errors in covariates, it will not engender biases for estimators ofparametric and nonparametric varying-coe?cient components obtained by semi-parametric profile least-squared methods. It only increases the correspondingasymptotic variances of those estimators. This tells us that SVCPL models havemany advantages of linear characteristics.