Efficient regression methods for flexible high dimensional models with heavy-tailed error

Technological advances have made high-dimensional data ubiquitous in many scientific fields. Driven by applications in fields such as biology, public health, genomics, medicine, social sciences, economy and finance, statisticians have made tremendous contributions to the development of high-dimensional statistical methods and theory. Existing works for high dimensional regression are mostly based on least squares approach and they mainly focus on light-tailed data. However, in real applications, heavy-tailed distribution commonly exists in high-dimensional data, which has been confirmed by many studies. Meanwhile, many methods were developed in the context of linear model, which restricts the applicability of the method. The overall target of this research project is to develop valid and efficient regression methods that utilize flexible semiparametric regression approaches to handle heavy-tailed high dimensional data. In the first project of this proposal, we study the convoluted rank regression for high dimensional partially linear additive model. We study the rate of convergence of oracle estimators and variable selection consistency for nonconvex penalized estimators. We also propose methods for tuning parameter selection. In the second project, we focus on sparse convoluted rank regression with varying coefficient structure. Rates of convergence for oracle estimator and L1 penalized estimator are investigated, and we also study the variable selection consistency for nonconvex penalized estimator. In both projects, we impose no condition on any moments of error and establish the theory. We will develop efficient algorithms for our method, and we will conduct simulation studies and apply the proposed methods to analyze some high-dimensional datasets concerning genomics, climate study, econometrics and finance.