High-dimensional data offers researchers increased ability to find useful factors in predicting a response. However, determination of the most important factors requires careful selection of the explanatory variables. In order to tackle this challenge, much work has been done on single or grouped variable selection under the penalized regression framework. Although the topic of variable selection has been extensively studied under the parametric framework, its applications to more flexible nonparametric models are yet to be explored.
In order to implement the variable selection in nonparametric additive models, I introduce and study two nonconvex selection methods under the penalized regression framework, namely the group MCP and the adaptive group LASSO, aiming at improvements on the selection performances of the more widely known group LASSO method in such models. One major part of the dissertation focuses on the theoretical properties of the group MCP and the adaptive group LASSO. I derive their selection and estimation properties. The application of the presently proposed methods to nonparametric additive models are further examined using simulation. Their applications to areas such as the economics and genomics are presented as well. Under both the simulation studies and data applications, the group MCP and the adaptive group LASSO have shown their advantages over the more traditionally used group LASSO method.
For the proposed adaptive group LASSO that uses the newly proposed weights, whose recursive application is therefore never studied before, I also derive its theoretical properties under a very general framework. Simulation studies under linear regression are included.
In addition to the theoretical and empirical investigations, throughout the dissertation, several other important issues have been briefly discussed, including the computing algorithms and different ways of selecting tuning parameters.