CAREER: High Dimensional Variable Selection and Risk Properties

高维变量选择在当代统计建模、学习和科学发现中起着举足轻重的作用。文献中长期存在的理论问题包括如何处理具有一般惩罚的高维正则化方法，惩罚函数的作用是什么，以及如何表征变量选择过程的最优性。研究者提出了四个相互关联的研究课题。首先，研究者研究了具有一般惩罚的惩罚似然方法，该方法被广泛应用于同时选择重要变量并估计其在高维统计推断中的影响，其中维数可以远大于样本量。其次，各种情况下的高维变量选择惩罚似然方法，包括惩罚经验风险和狩猎的相互作用进行了研究。第三，研究者提出了新的模型选择原则时，模型可能是错误的，并研究了各种正则化方法的鲁棒性下的模型错误指定的高维变量选择。第四，在惩罚最小二乘和惩罚似然的背景下，进一步研究了各种高维正则化方法的风险性质和最优性。现在，从基因组学和健康科学到经济学，金融学和机器学习，大量数据集的分析通常出现在科学，工程和人文科学的各个领域。高维数据分析对统计理论、方法和实现提出了许多挑战，这些挑战在较小规模的研究中不存在。这个建议的一个主要目标是使理论和方法的贡献，高维变量选择和统计推断的重要和具有挑战性的主题。这些新的发展提供了统一和系统的理解，在高维的各种正则化方法，并允许科学家分析高维数据，提高效率，方便性和可解释性。拟议的工作被纳入国家的最先进的高维统计学习的新课程，并将有利于本科生，研究生和代表性不足的少数民族的培训和学习。拟议的高维变量选择工作不仅有助于更好地确定对公共卫生和市场风险等重要因素，而且还使各个领域的广泛科学家和研究人员受益。

High dimensional variable selection plays a pivotal role in contemporary statistical modeling, learning and scientific discoveries. Long-standing theoretical questions in the literature include how high dimensionality regularization methods with general penalties can handle, what the role of penalty functions is, and how to characterize the optimality of variable selection procedures. The investigator proposes to study four interrelated research topics. First, the investigator studies penalized likelihood methods with general penalties, which are widely applied for simultaneously selecting important variables and estimating their effects in high dimensional statistical inference, where the dimensionality can be much larger than sample size. Second, various contexts of high dimensional variable selection beyond penalized likelihood methods including penalized empirical risk and hunting for interactions are investigated. Third, the investigator proposes new principles for model selection when models are possibly misspecified and studies the robustness of various regularization methods for high dimensional variable selection under model misspecification. Fourth, the risk properties and optimality of various high dimensional regularization methods in the contexts of penalized least squares and penalized likelihood are further investigated.The analysis of vast data sets now commonly arises in diverse fields of sciences, engineering and humanities ranging from genomics and health sciences to economics, finance and machine learning. High dimensional data analysis poses numerous challenges to statistical theory, methods and implementations that are not present in smaller scale studies. A major goal of this proposal is to make theoretical and methodological contributions to the important and challenging topic of high dimensional variable selection and statistical inference. These new developments provide unified and systematic understandings of various regularization methods in high dimensions, and allow scientists to analyze high dimensional data with increased efficiency, expediency and interpretability. The proposed work is incorporated into new courses on the state-of-the-art high dimensional statistical learning, and will benefit the training and learning of undergraduates, graduate students, and underrepresented minorities. The proposed work on variable selection in high dimensions will not only help better identify factors that are important to, for example, public health and market risk, but also benefit a broad range of scientists and researchers in various fields.