两类缺失机制下若干统计推断问题的研究

Due to various reasons,missing data is a very common phenomenon in real applications. Without using appropriate methods to consider the missing data,it is likely to lead to a large deviation for data analysis, even lead to a wrong conclusion. Hence, it is very important, not only in theoretical significance but also in pratical application value, to develop the statistical methods for missing data. The project mainly to study the statistical inference of semiparametric dimensional reducion for high-dimensional models with data missing at random and estimation based on impution method with data not missing at random, the main contents are as follows: (1) We first develop the semiparametric dimensional estimation for distribution function with data missing at random, avoiding the curse of dimensionality problem of high dimensional nonparametric estimation, and also the newly proposed estimation has some robustness;(2) We develop the statistical inferences of the semiparametric dimensional estimation for the parameters in estimating equation and semi-parametric regression models with data missing at random, and establish the empirical likelihood theory of the new method.(3) We study the inferences based on impution for estimations of distribution function, quantile function and estimation equation with data not missing at random, develop the impution method and the corresponding theoretical system;(4) We study the dimensional reduction estimation for high dimensional nonparametric functions with data not missing at random, because the complexity of data not missing at random and the difficulty of high dimensional nonparameter estimation, this problem is challenging.

由于各种各样的原因，数据缺失在实际应用中是一个非常普遍的现象。如果对缺失数据不用恰当的方法考虑，很有可能会导致数据分析产生很大的偏差，甚至会导致错误的结论。因此对于缺失数据下统计方法的研究具有非常重要的理论意义和实际应用价值。本项目主要研究随机缺失下高维模型的半参降维和非随机缺失下基于插补方法的统计推断，研究内容主要包括：（1）研究随机缺失下分布函数的半参降维估计，避免高维非参估计的维数祸根问题，且得到的估计具有稳健性；（2）研究随机缺失下估计方程和半参模型中参数的半参降维的统计推断，建立经验似然统计推断理论；（3）研究非随机缺失下分布函数、分位数和估计方程的基于插补方法的统计推断，建立相应的插补方法和理论体系；（4）研究非随机缺失下高维非参估计的降维问题，由于非随机缺失数据本身的复杂性和高维非参统计的本质困难，该问题的研究具有挑战性。

对缺失数据统计方法和理论的研究是统计界的研究热点问题之一，本项目主要研究随机缺失和非随机缺失下若干问题的统计推断。经过项目组三年的研究工作，我们得到了一系列的研究结果。具体的讲，在随机缺失机制下，建立了线性模型和非线性模型中的参数基于分位数方法的统计推断理论，建立了条件估计方程中参数的光滑最小距离估计的方法和理论，得到了分布函数半参降维的方法和理论。对非随机缺失下高维估计方程中参数的估计，提出了基于SIR（Sampling Importance Resampling）算法的插补估计方程，得到了参数的极大经验似然估计。项目基本按照计划书开展各项工作，达到了预期的目标。