Sure independence screening in generalized linear models with NP-dimensionality

Ultrahigh dimensional variable selection plays an increasingly important role in contemporary scientific discoveries and statistical research. Among others, Fan and Lv (2008) propose an independent screening framework by ranking the marginal correlations. They showed that the correlation ranking procedure possesses a sure independence screening property within the context of the linear model with Gaussian covariates and responses. In this paper, we propose a more general version of the independent learning with ranking the maximum marginal likelihood estimates or the maximum marginal likelihood itself in generalized linear models. We show that the proposed methods, with Fan and Lv (2008) as a very special case, also possess the sure screening property with vanishing false selection rate. The conditions under which that the independence learning possesses a sure screening is surprisingly simple. This justifies the applicability of such a simple method in a wide spectrum. We quantify explicitly the extent to which the dimensionality can be reduced by independence screening, which depends on the interactions of the covariance matrix of covariates and true parameters. Simulation studies are used to illustrate the utility of the proposed approaches. In addition, we � Supported in part by Grant NSF grants DMS-0714554 and DMS-0704337. The bulk of the work was conducted when Rui Song was a postdoctoral research fellow at Princeton University. The authors would like to thank the associate editor and two referees for their constructive comments that improve the presentation and the results of the paper. AMS 2000 subject classifications: Primary 68Q32, 62J12; secondary 62E99, 60F10