Tuning Parameter Selection Consistency in an Ultrahigh Dimensional Setup: A Comment on the Sure Independence Screening Rule

We congratulate Professors Fan and Lv for a thought-provoking paper, which provides us deep understanding about variable selection in an ultrahigh dimensional setup. We would like to supply our comments as follows. The important work of Breiman (1996) and Tibshirani (1996) demonstrated clearly that shrinkage estimation is a promising solution for variable selection. The first paper on the asymptotic results of the lasso appeared in Knight and Fu (2000). However, the important question regarding whether those shrinkage methods are consistent in model selection (Shao, 1997) was not clear. In a seminal paper, Fan and Li (2001) developed SCAD and, more importantly, introduced a general theoretical framework to understand the asymptotic behavior of various shrinkage methods. As a consequence, Fan and Li (2001) is also partially responsible for the recent development of the adaptive LASSO methods (Zou, 2006; Wang et al., 2007a; Zhang and Lu, 2007). Note that the oracle properties defined in Fan and Li (2001) depend on an appropriate selection of tuning parameters, for which prediction based criteria such as GCV have been commonly used in practice. Nevertheless, Leng et al. (2006) and Wang et al. (2007b) showed that this practice leads to seriously overfitted model. For model selection consistency, a BIC-type criterion is a justifiable alternative. Results were established for SCAD (Wang et al., 2007b) and adaptive LASSO (Wang and Leng, 2007) with a fixed dimension, and also for these two methods with diverging model dimensions (Wang et al., 2008)