Non-convex penalized estimation in high-dimensional models with single-index structure

As promising alternatives to the LASSO, non-convex penalized methods, such as the SCAD and the minimax concave penalty method, produce asymptotically unbiased shrinkage estimates. By adopting non-convex penalties, in this paper we investigate uniformly variable selection and shrinkage estimation for several parametric and semi-parametric models with single-index structure. The new method does not need to estimate the involved nonparametric transformation or link function. The resulting estimators enjoy the oracle property even in the “large pp, small nn” scenario. The theoretical results for linear models are in parallel extended to general single-index models with no distribution constraint for the error at the cost of mild conditions on the predictors. Simulation studies are carried out to examine the performance of the proposed method and a real data analysis is also presented for illustration.