Statistical Inference For Varying Coefficient Partially Nonlinear Model With Missing Covariates

Parametric models have the advantages of easy interpretation,which have been paid much attentions.However,if assumed parameter model is not correct,the results may be doubtful.Therefore,non-parametric methods are more often used to estimate the intrinsic function to reduce the bias of the model.The non-parametric model is flexible because it does not require the model’s structure.But when the dimension of variables is high,the problem of dimension curse will arise.Therefore,Scholars pay more and more attention to the semi-parametric model between parametric model and non-parametric model.The semi-parametric model contains both parametric components and non-parametric components.Therefore,the model not only retains the advantages of easy interpretation of the parameter model,but also has a wide range of adaptability.The varying coefficient partial-ly nonlinear model studied in this paper is a widely used semi-parametric model.Linear model,partially linear model and varying coefficient model are special cases of this model.In addition,missing data is often encountered in practical applications.For example,in the process of questionnaire survey,people will not answer questions concerning personal privacy,such as annual salary,marital history,age and so on.This leads to missing data.Standard statistical methods can not accurately estimate data,and often lead to deviations.Therefore,How to deal with missing data has become one of the hot topics in modern statistical analysis.Many scholars have studied the problem of missing data and proposed many methods.Among them,the method of inverse probability weighted is more popular.In this paper,we study the statistical inference for varying coefficient partially nonlinear model with missing covariates based on inverse probability weighted method.This paper is divided into five chapters.In chapter 1,we first introduce the basic knowledge related to varying coefficient partially nonlinear model,missing data,empirical likelihood and the current research status.Chapter two,chapter three and chapter four are the main work of this paper,chapter five is the sum-mary of this paper and the prospect of the future.In chapter 2,we give the least square estimation of the varying coefficient partially nonlinear model with missing covariates.We mainly give inverse probability weighted least squares estimation of parametric componentβand non-parametric componentsθ（u）when selection probabilityπ_iis unknown,which are denoted by?βand?θ（u）,respectively.It is also proved that the asymptotic normal result of parameter estimation is:^√n（?β-β）^D→N(0,Σ₁^-1Σ₂Σ₁^-1),The asymptotic normal result of nonparameter estimation is:^√nh[?θ（u）-θ（u）-₂¹h²μ₂θ^′′（u）]^D→N（0,Σ₃）,In the third chapter,we mainly give the empirical likelihood inference of the para-metric component and the non-parametric component in varying coefficient partial-ly nonlinear model with missing covariates.By constructing the estimated equation ofβandθ（u）,we obtain the empirical likelihood ratio statistic and prove the s-tatistics’asymptotic distribution is a standard chi-square distribution under mild conditions,so that the confidence intervals ofβandθ（u）are conveniently obtained without constructing the pivot statistics.Specifically,the confidence domain of the parameter is:C_α（β）={β:?L（β）≤χ²_1-α（p）}.the confidence domain of the nonparameter is:?（α）={θ（u）∈R^q:?l（θ（u））≤χ²_1-α（q）}In addition,in the fourth chapter,we also gave numerical simulations and real data analysis.From numerical simulations,we use the proposed inverse probabil-ity weighted least squares method and empirical likelihood method to carry out numerical experiments,and compare the results obtained by the two methods,and from the simulation results we can conclude that the empirical likelihood method proposed in this paper performs always better than the NA method for both the parameter components and the nonparametric function.Further more,The ex-ample analysis also verifies the feasibility of our method.In the fifth chapter,we summarize the main contents of the paper and the possible research work in the future.