Lasso-type Approach For Variable Selection In Single Index Model

In this thesis, we consider variable selection in single index models. The thesis is dividedinto two parts. In the first part, we review the literature on variable selection methods and es-timate methods for single index models in the last twenty years. In order to deal with variableselection, we reference to a non-concave penalty function, since it has ability to select significantvariables and to estimate unknown regression coefficients simultaneously. We highlight recentdevelopments on simultaneous variable selection and estimation through the methods of leastabsolute shrinkage and selection operator (Lasso). In the second part, we propose a penalizedlocal linear smoothing method, LASSO-type, for estimation and variable selection under thesingle index models. The sMAVE method penalizes the index vector of the single models andthe sim-LASSO method penalizes the partial derivative of the single models, and thus can beconsidered an extension of the usual LASSO method. In this paper, all extension of the usualLASSO are defined as LASSO-type. We studied an equivalent form of the sim-LASSO method,sLASSO, for variable selection in single index models under certain conditions. At the sametime, we propose a new LASSO-type method, penalizes relative partial derivative, for estima-tion and variable selection under the single index models. We develop iterative algorithm forcomputing proposed LASSO-type methods. In the end, the proposed LASSO-type methods areillustrated in the analysis of simulation study and a real data set.