Variable selection in parametric and semiparametric models

Since the proposal of the least absolute shrinkage and selection operator (LASSO) (Tibshirani, 1996) and the smoothly clipped absolute deviation (SCAD) method (Fan and Li, 2001), there have been extensive developments on model selection based on penalized log-likelihood and the computational issues of solving these problems in linear models. There are relatively less papers discussing model selection problems in some special but important parametric and semiparametric models, such as linear mixed-effects model, partially varying-coefficient single-index model and single-index-coefficient regression model. We propose a two-stage model selection procedure for the linear mixed-effects model. The procedure consists of two steps: First, penalized restricted log-likelihood is used to select the random effects, and this is done by adopting a Newton-type algorithm. Next, the penalized log-likelihood is used to select the fixed effects via pathwise coordinate optimization to improve the computation efficiency. For the partially varying-coefficient single-index model, we propose a class of efficient penalized estimating equations, which combine the smoothly clipped absolute deviation (SCAD) penalty and a stepwise estimation method. We propose a new procedure for model structure determination for the single-index-coefficient regression models; that is, the penalized estimating equations (PEE) for variable selection that combines the "delete-one-component" method and the smoothly clipped absolute deviation penalty. The proposed PEE method can simultaneously identify significant variables of the index and estimate the nonzero coefficients of the index parameters. We further study a hypothesis test procedure for nonparametric index-coefficient functions. Monte Carlo simulation studies are conducted to assess the finite sample performance of the proposed methods. And real-life examples are analyzed using the proposed methods as an illustration.​Doctor of Philosophy (SPMS