基于广义SICA惩罚函数的高维数据参数估计与变量选取研究

With the rapid development of information technology, high-dimensional data is emerging in many fields such as management science, finance and economics, informatics and medicine, which attracts a lot of attention from scholars. The penalized method for high-dimensional data is a hot topic in recent years. Although there are many significant results in the literature, few attentions are focused on the structure principle of penalty functions, and there is no general framework for this aspect. To this end, we first build a theoretical system of generalized SICA penalized likelihood estimation based on the original idea of penalty functions and their conclusions, which reveals the essence of SICA, TLP, SELO and ArctanLASSO. We prove the asymptotical properties of the estimator, and try to search for the best penalty based on the given data. Secondly, we promote and stabilized the generalized SICA penalized estimator under the framework of Bootstrap, and the promoting estimator is proved to be consistent. Finally, we apply the proposed method to sparse classification, parameter estimation and variable selection of generalized linear model, network exploration on high dimensions and piecewise linear trend filtering of time series, and we hope the results can be more precise. The conclusion of this project is of great theoretical and practical significance, and provides an efficient tool for high-dimensional data and statistical learning.

随着信息技术的迅速发展，高维数据涌现于管理科学、金融经济、信息科学等众多领域，受到越来越多学者的关注。高维数据的惩罚方法是近几年的研究热点，虽然研究结论颇多，但是极少有学者考虑惩罚方法的构造原理，也并没有给出一般性的框架。为此，我们首先从惩罚方法的源头出发，综合前人的成果，建立高维数据下广义SICA惩罚似然估计方法的理论体系，揭示了SICA、TLP、SELO、ArctanLASSO等惩罚函数的构造原理，给出了估计量的渐近性质和有效的算法，并且试图寻找基于数据的最优罚函数；其次在Bootstrap框架下给出该方法的提升算法，并证明估计结果的相合性；最后将该方法运用于稀疏分类算法、广义线性模型的稀疏估计、高维数据的网络探测以及分段线性的时间序列趋势滤波算法，以期能够得到更为精确的估计。本项目的结论为高维数据和统计学习提供了有力的分析方法，有着重要的理论意义和实际意义。

参数估计和变量选取是统计学习理论中重要的研究方向。本项目针对变量较多的数据下进行了理论型和应用型研究，其中理论性研究包含了多来源异质性数据下广义线性模型Maximin似然比稀疏估计、混频时间序列的潜在因子分析和大规模数据下基于充分降维的重要性抽样方法，应用性研究涵盖金融和医学两个方向，主要包含银行杠杆决策的影响因子、晚期头颈癌患者肿瘤切除后90天的风险评估、老年女性口腔癌的预测因子等。本项目的研究为相关领域提供了有价值的理论基础和应用场景，具有一定的现实意义。