Dissertation
Dissertation > Mathematical sciences and chemical > Mathematics > Probability Theory and Mathematical Statistics > Mathematical Statistics > General mathematical statistics

Statistical Inference for Partially Linear Models with Missing Responses

Author LiYingHua
Tutor QinYongSong
School Guangxi Normal University
Course Probability Theory and Mathematical Statistics
Keywords partially linear model missing data fixed design point MAR missing mechanism confidence interval
CLC O212.1
Type Master's thesis
Year 2009
Downloads 39
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In practice, some data may be missing for various reasons such as unwillingness of some sampled units to supply the desired information, loss of information caused by uncontrollable factors, failure on the part of the investigator to gather correct information, and so forth. Missing data problem has gained more and more attention in practice. The statistical inference for the partially linear models with missing data which has extensive application background becomes a focus for study. In such circumstances, 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 important 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 imputation and semiparametric regression imputation. When the coverate data are missing at random in a partially linear model with random design points, Wang (Statistical estimation in partial linear 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 the parametric and nonparametric parts. When the response data are missing at random in a partially linear model with random design points, Wang et al. (Semiparametric regression analysis with missing response at random [J]. J Amer Statist Assoc, 2004, 99: 334-345) developed an empirical likelihood method to make inference for the mean of the response variable, and Wang and Sun (Estimation in partially linear models with missing responses at random [J]. J Multivariate Anal, 2007, 98: 1470-1493) developed semiparametric regression imputation and inverse probability weighted approaches to estimate the parametric and nonparametric parts. In Chapter 2 of this paper, the inference in a partially linear model with non-random design points and missing data is studied. We develop semiparametric regression imputation and inverse probability weighted approaches to estimate the parametric and nonparametric parts in a partially linear model with non-random design points. Asymptotic normality of the estimators is established, which is used to construct normal approximation based confidence intervals on the the parametric and nonparametric parts. In Chapter 3 of this paper, empirical likelihood (EL) ratio statistics on the parametric and non-parametric parts in a partially linear model with non-random design points are constructed based on the inverse probability weighted imputation approach, which asymptotically have chi-squared distributions. These results are used to obtain EL based confidence intervals(regions) on the the parametric and nonparametric parts without adjustment, which can improve the accuracy of the confidence intervals(regions). Note that the EL statistic which is constructed based on ’complete sample’ after regression imputation has a limiting distribution of a weighted sum of chi-squared variables, see Wang et al. (Semiparametric regression analysis with missing response at random[J]. J Amer Statist Assoc, 2004, 99: 334-345), Wang and Rao (Empirical likelihood-based inference in linear models with missing data[J]. Scandinavian Journal of Statistics, 2002, 29(2): 563-576; Empirical likelihood-based inference under imputation for missing response data[J]. Ann Statist, 2002, 30(3): 896-924). So they need to use an adjusted EL to obtain a confidence intervals(regions) on the parametric and nonparametric parts based on the ’complete sample’ 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. The statistical inference for partially linear models with non-random design points is an important research topic theoretically and practically, and the inference in a partially linear model with non-random design points and missing data is an untouched issue. We study this issue in this paper.2. The inference in a partially linear model with non-random design points and missing data is studied for the first time. We develop semiparametric regression imputation and inverse probability weighted approaches to estimate the parametric and nonparametric parts in a partially linear model. Asymptotic normality of the estimators is established, which is used to construct normal approximation based confidence intervals(regions) on the parametric and nonparametric parts.3. In studying the construction of confidence intervals(regions) for the parametric and nonparametric parts in a partially linear model with non-random design points, 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 with non-random design points are constructed, which asymptotically have chi-squared distributions. These results are used to obtain EL based confidence intervals (regions) on the parametric and nonparametric parts without adjustment, which can improve the accuracy of the confidence intervals(regions).

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