基于非凸惩罚似然法的混合回归模型稳健性推断

The classical mixture regression model with maximum likelihood estimate (MLE) fails to work in the presence of only a few severe outliers. Therefore, developing robust mixture regression models becomes necessary and pressing for current multivariate statistics. In this project, we mainly propose to extend the robust mixture modeling (RMM) approach, which successfully uses a mean-shift formulation coupled with nonconvex penalized likelihood, to mixture regression models, mixture generalized linear models，and semiparametric mixture regression models with varying mixing proportions. These three methods are expected to achieve simultaneous outlier detection and robust parameter estimation. Efficient iterative thresholding-embedded EM algorithms will be developed, and some theoretical properties such as breakdown point and selection consistency of outlier detection will be investigated in these three methods. Furthermore, we propose to identify the predictor effects in all components and conduct variable selection for mixture regression models using lasso objective function with equality constraints. For this purpose, a Bregman coordinate descent algorithm (BCDA) will be developed. Prospectively, this project will enhance the researches on mixture regression model in the cases whether the data are contaminated by outliers and broaden its applications.

当数据中出现离群值时，传统混合回归模型的极大似然法不能进行准确的参数估计。所以，混合回归模型的稳健推断研究已经成为迫切而必要的课题。本项目拟将基于非凸惩罚似然法的稳健混合模型延伸应用到混合回归模型、混合广义线性模型和半参数的混合回归模型。可以预期的是，这三种稳健混合回归模型能同时进行离群值检测和稳健推断。本项目将为这三种稳健混合回归模型分别设计出迭代的阈值嵌入式EM算法，并深入研究其鲁棒性和离群值检测的选择一致性等理论性质。此外，本项目拟引入基于等式约束的lasso目标函数来研究混合回归模型的自变量效应分类和变量选择，并为此方法设计出BCDA算法。本项目研究有望进一步丰富和开拓混合回归模型的研究与应用。

当数据中出现离群值时，传统混合回归模型的极大似然法不能进行准确的参数估计。所以，混合回归模型的稳健推断研究已经成为迫切而必要的课题。本项目成功地将基于非凸惩罚似然法的稳健回归模型和混合模型方法延伸应用到混合回归模型和广义线性模型以进行参数估计和变量选择。这两种新的稳健模型能同时进行离群值检测和稳健推断。本项目还为稳健混合回归模型的参数估计和变量选择以及稳健广义线性模型的参数估计分别设计出迭代的阈值嵌入式EM算法，并深入研究其鲁棒性和离群值检测等理论性质。此外，本项目拟引入基于等式约束的lasso目标函数来研究混合回归模型的自变量效应分类和变量选择，并为此方法设计出BCDA算法。本项目的研究成功地用于各领域的实际应用，如将基于非凸惩罚似然法的稳健混合回归模型用于研究月相和温度对棉红铃虫成虫种群动态的影响，用基于非凸惩罚似然法的稳健Logistic回归模型研究网红经济对消费者购买行为影响因素等。这些研究成果有望进一步应用于农业、环境保护、经济、商业、市场、社会科学等领域，具有非常广阔的应用前景。