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. In addition, we establish an
exponential inequality for the quasi-maximum likelihood estimator which is
useful for high-dimensional statistical learning.