LARGE SAMPLE PROPERTIES OF THE SCAD-PENALIZED MAXIMUM LIKELIHOOD ESTIMATION ON HIGH DIMENSIONS

In this paper, we study large sample properties of smoothly clipped absolute deviation (SCAD) penalized maximum likelihood estimation for high-dimensional parameters. First, we prove that the oracle maximum likelihood estimator (MLE) asymptotically becomes a local maximizer of the SCAD-penalized log-likelihood, even when the number of parameters is much larger than the sample size; the oracle MLE is an ideal non-penalized MLE obtained by deleting all irrelevant parameters in advance. Second, we prove that if the log-likelihood is strictly concave, the oracle MLE asymptotically becomes the global maximizer of the SCAD-penalized log-likelihood with a diverging number of parameters that is less than the sample size. Various numerical experiments on simulated data sets are presented to verify the theoretical results, and two data examples are analyzed.