Thresholding-based Iterative Selection Procedures for Generalized Linear Models

High-dimensional correlated data pose challenges in model selection and predictive learning. In this paper, we derive an iterative thresholding technique for generalized linear models (GLMs) with possibly nonorthogonal designs. We propose a family of $\Theta$-estimators which are associated with penalized likelihoods and can be computed by thresholding-based iterative procedures. It can also be used to robustify GLMs and extend the canonical $M$-estimators. In particular, the thresholding technique applies to a fusion of the $l_0$-penalty and ridge-penalty which has outstanding performance in model selection and prediction. A novel selective cross-validation (SCV) scheme is also proposed for nonconvexity parameter tuning. Real microarray data are analyzed to illustrate the proposed methodology. Our results extend to grouped GLMs.