Empirical Likelihood Inferences For Two Classes Of Statistical Models With Missing Responses

Data missing is the frequent phenomenon in real life, such as reliability of life testing, marketresearch of follow-up tests in medicine which lies in a large number of missing data. There aremany reasons for missing data ,which can be divided into two categories of human factors andobjective factors,if people are unwilling to answer privacy issues, (income and age, etc.). somedata can not be observed due to technical reasons or give up sampling because of cost considera-tions, etc.Therefore, the practical application of missing data will arouse more and more concern.In such circumstances, the usual inferential procedures for complete data sets cannot be applieddirectly. 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'completesample'set and then apply standard statistical methods. Statistical inference for missing data is animportant research field (e.g. Little and Rubin, Statistical Analysis with Missing Data[M], NewYork: John Wiley and Sons 2002). In the study of the regression models with missing data, com-monly used imputation approaches include linear regression imputation, nonparametric regressionimputation and semiparametric regression imputation. When the coverate data are missing at ran-dom in a partially linear model with random design points, Wang (Statistical estimation in partiallinear models with covariate data missing at random [J]. Ann Inst Stat Math, 2009, 61: 47-84)developed a model calibration approach and a weighting approach to define the estimators of theparametric and nonparametric parts. When the response data are missing at random in a partiallylinear model with random design points, Wang et al. (Semiparametric regression analysis withmissing response at random [J]. J Amer Statist Assoc, 2004, 99: 334-345) developed an empir-ical likelihood method to make inference for the mean of the response variable,Wang and Rao(Empirical likelihood-based inference under imputation for missing response data[J]. Ann Statist,2002, 30(3):896-924) discuss the construction of confidence intervals(regions) for responses vari-able mean in a nonparametric regression model with missing data, and Wang and Sun (Estimationin partially linear models with missing responses at random [J]. J Multivariate Anal, 2007, 98:1470-1493) developed semiparametric regression imputation and inverse probability weighted ap-proaches to estimate the parametric and nonparametric parts.In Chapter 2 of this paper, the estimation and asymptotic normality of the estimators in a partially linear model with random design point and missing responses is studied. We obtain asfollows results.(1)We develop a new inverse probability weighted approaches to estimate the parametric partsin a partially linear model , prove asymptotic normality of the estimators and obtain asymptoticvariance more simple than Wang and Sun (Estimation in partially linear models with missing re-sponses at random[J]. J Multivariate Anal, 2007, 98: 1470-1493).Asymptotic normality of theparametric estimators is established, which is used to construct normal approximation based con-fidence intervals on the the parametric parts.(2)We develop a new inverse probability weighted approaches to estimate the nonparamet-ric parts in a partially linear model for the first time ,and prove asymptotic normality of the es-timators.Asymptotic normality of the nonparametric estimators is established, which is used toconstruct normal approximation based confidence intervals on the nonparametric parts.(3)We weaken conditions and broaden the applicable scope of the approach and models com-bined with Wang and Sun(Estimation in partially linear models with missing responses at ran-dom[J]. J Multivariate Anal, 2007, 98: 1470-1493).In Chapter 3 of this paper, empirical likelihood (EL) ratio statistics on the parametric and non-parametric parts in a partially linear model for the first time are constructed based on the inverseprobability weighted imputation approach, which asymptotically have chi-squared distributions.Accordingly, we construct empirical likelihood confidence intervals for the parametric and non-parametric parts .In Chapter 4 of this paper,empirical likelihood (EL) ratio statistics on the responses variablemean in a nonparametric regression model for the first time are constructed based on the inverseprobability weighted imputation approach, which asymptotically have chi-squared distributions.Accordingly, we construct a empirical likelihood confidence interval for the responses variablemean.These results are used to obtain EL based confidence intervals(regions) on the the parametric, nonparametric parts and the responses variable mean without adjustment, which can improvethe accuracy of the confidence intervals(regions). Note that the EL statistic which is constructedbased on'complete sample'after regression imputation has a limiting distribution of a weightedsum of chi-squared variables, see Wang et al. (Semiparametric regression analysis with miss-ing response at random[J]. J Amer Statist Assoc, 2004, 99: 334-345), Wang and Rao (Empiricallikelihood-based inference in linear models with missing data[J]. Scandinavian Journal of Statis-tics, 2002, 29(2): 563-576; Empirical likelihood-based inference under imputation for missingresponse data[J]. Ann Statist, 2002, 30(3): 896-924). So they need to use an adjusted EL to obtaina confidence intervals(regions) on the parametric and nonparametric parts based on the'completesample'after regression imputation, in which the adjustment coefficient needs to be estimated.This would lead to a loss of the accuracy of the confidence intervals(regions). Here we summary some new findings in this paper.1. In studying the statistical inference for partially linear models with missing responses atrandom , We develop a new inverse probability weighted approaches to estimate the parametricand nonparametric parts in a partially linear model.Asymptotic normality of the estimators is es-tablished, which is used to construct normal approximation based confidence intervals(regions) onthe parametric and nonparametric parts,as the same time of weakening conditions and broaden-ing the applicable scope of the approach and models combined with Wang and Sun(Estimationin partially linear models with missing responses at random[J]. J Multivariate Anal, 2007, 98:1470-1493).2. In discussing the construction of confidence intervals(regions) for the parametric and non-parametric parts in a partially linear model with missing responses at random for the first time,we use the inverse probability weighted imputation approach. Based on this imputation approach,EL ratio statistics on the parametric and nonparametric parts in a partially linear model are con-structed, which asymptotically have chi-squared distributions. These results are used to obtain ELbased confidence intervals(regions) on the parametric and nonparametric parts without adjustment,which can improve the accuracy of the confidence intervals(regions).3. In discussing the construction of confidence intervals(regions) for responses variable meanin a nonparametric regression model with missing data for the first time, we use the inverse prob-ability weighted imputation approach. Based on this imputation approach, EL ratio statistics onthe responses variable mean in a nonparametric regression model are constructed, which asymp-totically have chi-squared distributions. The result are used to obtain EL based confidence inter-vals(regions) on the parametric and nonparametric parts without adjustment, which can improvethe accuracy of the confidence intervals(regions).