不完全数据下分位数回归模型的经验似然推断

Quantile regression is very robust against outliers and can describe the entire conditional distribution of the response variable given the covariates. However, in practice direct application of quantile regression is hindered, because incomplete data often occurs, even inevitable. To the best of our knowledge, inference for the parameter of a quatile regression model with censored responses and missing partially covariates, has not been developed. Suppose that the response and the censoring variable are independent given the observed covariates and the missing covariates are missing at random, we intend to construct a class of empirical likelihood-based robust inference functions and give the estimator of the regression parameter as well as its asymptotic properties. By using empirical likelihood-based robust inference functions, we can get the chi-square test and the corresponding confidence region of the regression parameter. On one hand, empirical likelihood-based robust inference functions are robust to outliers. On the other hand, empirical likelihood-based robust inference functions can handle the problem of censored responses and missing partially covariates and thus improve the inference efficiency. We will extend the theory of empirical likelihood-based robust inference functions to longitudinal data and repeated measurements with censored responses and missing partially covariates.

分位数回归不但减弱了异常值对推断的影响，具有稳健性，而且还能在给定协变量的条件下，完整地刻画响应变量的条件分布。在实际应用中，数据不完全的情况经常发生，甚至是不可避免的。这给实际工作者使用分位数回归方法造成了很大的困难。据我们所知，在响应删失且部分协变量数据缺失的情况下，关于分位数回归的研究迄今为止还是空白。我们打算在响应变量和删失变量条件独立(给定完全观测的协变量)的假设下及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。此推断函数一方面减弱了异常值对统计推断的影响，具有稳健性；另一方面，克服了响应删失且部分协变量数据缺失的影响，提高了推断效率。我们将推广不完全数据下基于经验似然的分位数回归理论，使其可以处理响应删失且部分协变量数据缺失的纵向数据和重复测量数据。

分位数回归以其稳健性的优势在经济学，社会科学，生物和医学领域得到广泛应用。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，直接应用分位数回归方法，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分位数回归，复合分位数回归，变系数分位数回归和加速失效时间模型，得到了高效且稳健的回归参数估计及其渐近性质。