Shrinkage Tuning Parameter Selection with a Diverging Number of Parameters

Contemporary statistical research frequently deals with problems involving a diverging number of parameters. For those problems, various shrinkage methods (e.g., LASSO, SCAD, etc) are found particularly useful for the purpose of variable selection (Fan and Peng, 2004; Huang et al., 2007b). Nevertheless, the desirable performances of those shrinkage methods heavily hinge on an appropriate selection of the tuning parameters. With a fixed predictor dimension, Wang et al. (2007b) and Wang and Leng (2007) demonstrated that the tuning parameters selected by a BIC-type criterion can identify the true model consistently. In this work, similar results are further extended to the situation with a diverging number of parameters for both unpenalized and penalized estimators (Fan and Peng, 2004; Huang et al., 2007b). Consequently, our theoretical results further enlarge not only the applicable scope of the traditional BIC-type criteria but also that of those shrinkage estimation methods (Tibshirani, 1996; Huang et al., 2007b; Fan and Li, 2001; Fan and Peng, 2004).