The Statistical Inference For Additive Partial Linear Models With Missing Covariates

In the models,people prefer to study the parameter model,mainly be-cause the parameter model is simple,interpretive,and effective.However,if the model assumptions which we give are inaccurate,there will be erroneous results.Therefore,in order to reduce the deviation of the parameter model,non-parametric model is increasingly favored by people because of its flexibility.However,when the covariates dimension is,relatively high,there will be a problem of "curse of dimensionality".Therefore,the semi-parametric models which between the para-metric and non-parametric model are getting more and more attention.It is not only mainly because the semi-parametric models have the interpretability of the parameter model,but also they inherit the flexibility of the non-parametric model.The additive partially linear model is one commonly used model among the semi-parametric models.In real life,many areas such as market research,many surveyed people do not want to disclose their information,which may result in missing data during the time of the survey.In this case,the standard statistical methods can not be able to analyze the incomplete data.Therefore,many scholars have studied the problem on how to use the observed data to get the statistical inferences.For the method of dealing with missing data,the first thing people can think of is to remove the data that is not observed,and use the data that can be observed for statistical analysis.However,this method has the disadvantage that the statistical analysis results are biased.Therefore,another more effective method,the inverse probability weight-ing(IPW)method,is being popularly used by people.This article is based on the method of inverse probability weighting in the absence of covariates.This paper is divided into four chapters.In the first chapter,we first intro-duce the basic knowledge and the current research status of missing data,quantile regression,empirical likelihood,and additive partially linear model.Chapters 2 and 3 are the main work of this article-in the fourth part,we give the summa-ry of this article and the future outlook.The second chapter is based on Ben Sherwood's(2016)[5]for the parametersU single quantile estimation of the ad-ditive partial linear model with missing covariates,we propose the parameter ?weighted quantile average estimators under probability ?i is known,with parame-ter structure and non-parametric structure under,respectively,that is:(?).This arti-cle also prove the asymptotic normal distribution of the parameter:(?)In addition,in the fourth part of the second chapter we also gave numerical sim-ulations and data analysis.From numerical simulations we can found that our proposed method is the best among the composite quantile(WCQR)and quantile(WQR)and least squares(WLS).In the third chapter,we give the inverse proba-bility weighted least square estimator and the empirical likelihood inference for the parameters in the partially additive partially linear model with missing covariates.The asymptotic normal distribution of the inverse probability weighted least square estimator is proved,that is:(?)According to its asymptotic normality,the corresponding confidence interval can be given,but when constructing the confidence intervals,?1 and ?2 need to be es-timated in advance,so the confidence interval is not accurate.So we consider the empirical likelihood inference of ?.By constructing the estimated equation of ?,we obtain the empirical likelihood ratio statistic and prove the statistics' asymptotic distribution is a standard chi-square distribution under mild conditions,so that the confidence intervals of the parameter ? are conveniently obtained without con-structing the pivot statistics.Specifically,the confidence domain of the parameter is:(?)...