Statistical Inference For Varying-coefficient Partially Linear Models With Missing Responses

Varying-coefficient partially linear models extensively cover some important semiparametric models such as partially linear models and so on. Their advantages lie in their good combina-tion of three-folds. On one hand, they have the merits of linear model which are prone to easily interpretation, constructing estimation and tests. On the other hand, they display robust and flexi-ble virtues as for nonparametric models. Furthermore, their varying-coefficient part may expound an interaction between covariates, dynamic changs (e.g. when varying-coefficients are related to time). Besides, they allow more flexible function forms, and reduction of the dimension of data and so on. Therefore they are widely applied in finance, economics, biomedicine and some other fields.In practice, such as market research of follow-up tests, in medicine, reliability of life testing, some data may be missing for reasons such as unwillingness of some sampled units to supply the desired information, loss of information caused by two categories of human factors and objective factors, so missing data problem has gained more and more attention in practice. In such cir-cumstances, the usual inferential procedures for complete data sets cannot be applied directly. It needs to do some treatments on data before we can use usual statistical approaches. A common method is to impute values for each missing response in order to obtain a'complete sample'set and then apply standard statistical methods. Statistical inference for missing data is an impor-tant research field (e.g. Little and Rubin, Statistical Analysis with Missing Data[M], New York: John Wiley and Sons 2002). In the study of the regression models with missing data, commonly used imputation approaches include linear regression imputation, nonparametric regression im-putation and semiparametric regression imputation. In this paper, we develop inverse probability weighted approaches to make inference for the mean of the responses and the parameters in a varying-coefficient partially linear model with missing responses.This thesis is divided into three chapters. Chapter one contains introduction and relevant literature reviews.In Chapter two, we develop inverse probability weighted approaches to estimate the mean of the responses and the parameters in a varying-coefficient partially linear model. Asymptotic normality of the estimators is established, which is used to construct normal approximation based confidence intervals (regions) on the mean of the responses and the parameters. In Chapter three, based on the inverse probability weighted imputation approach, the EL ratio statistics on the mean of the responses and the parameters in a varying-coefficient partially linear model with random design points are constructed, which asymptotically have chi-squared distribu-tions. These results are used to obtain EL based confidence intervals (regions) without adjustment, which can improve the accuracy of the confidence intervals (regions).Here we summary some new findings in this paper.1. We develop a new inverse probability weighted approach to estimate the mean of the responses and the parameters in a varying-coefficient partially linear model. Asymptotic normality of the estimators is established, which is used to construct normal approximation based confidence intervals (regions) on the mean of the responses and the parameters.2. In studying the construction of confidence intervals (regions) for the mean of the responses and the parameters in a varying-coefficient partially linear model with random design points, we use the inverse probability weighted imputation approach for the first time. Based on this imputa-tion approach, EL ratio statistics on the mean and the parameters in a varying-coefficient partially linear model with random design points are constructed, which asymptotically have chi-squared distributions. These results are used to obtain EL based confidence intervals (regions) on the mean of the responses and the parameters without adjustment, which can improve the accuracy of the confidence intervals (regions).