缺失数据下基于经验似然的稳健推断函数

Rank regression is a very efficient and robust nonparametric method. However, in practice, direct application of rank regression is hindered, because missing data often occurs, even inevitable. To the best of our knowledge, inference for the parameter of a rank regression model with missing responses or partially covariates, has not been developed. Suppose that the missing response and the missing covariates are missing at random , we intend to construct a class of empirical likelihood-based robust inference function and give the estimator of regression parameter as well as its asymptotic properties. By using empirical likelihood-based robust inference function, we can get the chi-square test and the corresponding confidence region of the regression parameter. On one hand, empirical likelihood-based robust inference function is rank-based and thus robust to outliers. On the other hand, empirical likelihood-based robust inference function can handle the problem of missing responses or partially covariates and thus improve the inference efficiency. We will extend the theory of empirical likelihood-based robust inference function to the longitudinal data and repeated measurements as well as the quantile regression. Finally, based on the R software platform, we will develop software packages to implement the proposed statistical methods and algorithms.

秩回归是一种高效且稳健的非参数方法。但在实际应用中，数据缺失的情况经常发生，甚至是不可避免的。这给实际工作者使用秩回归方法造成了很大的困难。据我们所知，在响应变量或部分协变量数据缺失的情况下，关于秩回归的研究迄今为止还是空白。我们打算在响应变量或部分协变量随机缺失机制下构造一类基于经验似然的稳健推断函数，并给出回归参数的估计及其渐近性质。通过使用基于经验似然的稳健推断函数，我们可以得到参数的卡方检验以及相应的置信域。基于经验似然的稳健推断函数一方面是基于秩的，减弱了异常值对统计推断的影响，具有稳健性；另一方面，克服了响应变量或部分协变量缺失的影响，提高了推断效率。我们将推广缺失数据下基于经验似然的秩回归理论，使其可以处理：(1)缺失响应变量或部分协变量的纵向数据和重复测量数据；(2)缺失响应变量或部分协变量的分位数回归模型。最后，我们将基于R软件平台，开发软件包实现本项目所提出的统计方法。

logistic疾病风险回归，分位数回归和秩回归都是稳健的回归方法。但在实际应用中，数据缺失的情况经常发生，甚至是不可避免的。如果忽略缺失数据，直接应用稳健的回归方法，将产生低效甚至有偏的推断。本项目在不完全数据下，构造了一类基于经验似然的稳健推断函数。基于经验似然的稳健推断函数减弱了异常值对统计推断的影响，具有稳健性；另一方面，克服了响应变量或部分协变量缺失的影响，提高了推断效率。我们将基于经验似然的稳健推断函数分别应用于logistic疾病风险回归，分位数回归和秩回归，得到了高效且稳健的回归参数估计及其渐进性质。