Variable selection for ultra-high-dimensional logistic models

We propose a variable selection procedure through the optimization of a nonconcave penalized likelihood for logistic regression models with the dimension of covariates p diverging in an exponential rate of n. We first establish the oracle property of the procedure under such ultra-high-dimensional setting. Our optimization algorithm combines some recent developments, including the concave convex procedure and the coordinate descent algorithm, in solving regularization problems. Through extensive simulations, we show the promise of the proposed procedure in various high-dimensional logistic regression settings. An application to gene expression data from a breast cancer study illustrates the use of the method.