Author's personal copy Profiled adaptive Elastic-Net procedure for partially linear models with high-dimensional covariates Baicheng Chen a , Yao Yu b , Hui Zou c , Hua Liang b,n a Department of Statistics, Shanghai University of Finance and Economics, Shanghai, China b Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY 14642, USA c Department of Statistics, University of Minnesota, Minneapolis, MN 55455, USA article info Article history: Received 8 July 2011 Accepted 15 February 2012 Available online 23 February 2012 Keywords: Adaptive regularization Elastic-Net High dimensionality Model selection Oracle property Presmoothing Semiparametric model Shrinkage methods abstract We study variable selection for partially linear models when the dimension of covariates diverges with the sample size. We combine the ideas of profiling and adaptive Elastic-Net. The resulting procedure has oracle properties and can handle collinearity well. A by-product is the uniform bound for the absolute difference between the profiled and original predictors. We further examine finite sample performance of the proposed procedure by simulation studies and analysis of a labor-market dataset for an illustration. & 2012 Elsevier B.V. All rights reserved. 1. Introduction Consider the partial linear models (PLM H¨ ardle et al., 2000) Y ¼ X T b þ gðZÞþ e, ð1:1Þ where X ¼ðx 1 , ... , x p Þ T and Z are the linear and nonparametric components, gðÞ is an unknown smooth function. We are interested in variable selection procedure for parametric components X when the dimension number p is large, which may depend upon the sample size. We propose to use the adaptive Elastic-Net (Zou and Zhang, 2009) for variable selection in the PLM using profile least squares approach to convert the partial linear models to the classical linear regression model. In the past decade, we have witnessed great progress in variable selection for a variety of models, since two elegant penalized based methods, the least absolute shrinkage and selection operator (LASSO) penalty (Tibshirani, 1996) and the smoothly clipped absolute deviation (SCAD) penalty (Fan and Li, 2001, 2002), had been proposed. A large body of penalized methods has been studied in the literature. See for example Zou and Hastie (2005), Meinshausen and B ¨ uhlmann (2006), Zou (2006, 2008), Zhao and Yu (2006), Huang et al. (2008, 2009), and van de Geer (2008). Recently, researchers have also considered applications of penalization methods in semiparametric and nonparametric models. For instance, Li and Liang (2008) for semiparametric models and Liang and Li (2009) for partially linear models with measurement errors. Huang et al. (2010) and Ravikumar et al. (2009) investigated high-dimensional nonparametric sparse additive models. Xie and Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jspi Journal of Statistical Planning and Inference 0378-3758/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2012.02.035 n Corresponding author. E-mail address: hliang@bst.rochester.edu (H. Liang). Journal of Statistical Planning and Inference 142 (2012) 1733–1745