Collaborative Research: Penalization Methods for Screening, Variable Selection and Dimension Reduction in High-Dimensional Regression via Multiple Index Models

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The project aims to develop effective penalization methods for screening, dimension reduction, and variable selection in high dimensional regression. The investigators focus mainly on multiple index models, because this type of models combines the strengths of linear and nonparametric regression while avoiding their drawbacks. A novel penalization approach is employed for model fitting, which regularizes both the parametric and nonparametric components of a multiple index model. A pilot study shows that this approach is more advantageous than other existing ones. When facing ultra-high dimensionality, the investigators use a forward variable screening procedure to reduce the dimension to a manageable size before applying the proposed penalization. The investigators plan to study the theoretical properties of this approach and develop fast and efficient computing algorithms for its implementation. The proposed approach is further extended to applications involving categorical responses or random effects.Advances in science and technology have led to an explosive growth of massive data across a variety of areas such as bioinformatics, climate research, internet, etc. Traditional statistical methods for clustering, regression and classification become ineffective when dealing with a large number of variables. Lately, a tremendous amount of research effort has been dedicated to the development of statistical methods such as dimension reduction and variable selection for analyzing this type of massive data. The investigators join the effort by proposing a novel penalization approach and developing efficient computing algorithms. The results from this project not only advance statistical research but also help other scientists and researchers better understand and analyze their massive data and hence enhance their scientific discovery.