Nonconcave penalized likelihood with NP-dimensionality

Jianqing Fan Jinchi Lv

Statistics Theory and Methods mathscidoc:1912.43271

IEEE Transactions on Information Theory, 57, (8), 5467-5484, 2011.7
Penalized likelihood methods are fundamental to ultrahigh dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models, such methods possess model selection consistency with oracle properties even for dimensionality of nonpolynomial (NP) order of sample size, for a class of penalized likelihood approaches using folded-concave penalty functions, which were introduced to ameliorate the bias problems of convex penalty functions. This fills a long-standing gap in the literature where the dimensionality is allowed to grow slowly with the sample size. Our results are also applicable to penalized likelihood with the <i>L</i> <sub>1</sub> -penalty, which is a convex function at the boundary of the class of folded-concave penalty functions under consideration. The coordinate optimization is implemented for finding the
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  title={Nonconcave penalized likelihood with NP-dimensionality},
  author={Jianqing Fan, and Jinchi Lv},
  booktitle={IEEE Transactions on Information Theory},
Jianqing Fan, and Jinchi Lv. Nonconcave penalized likelihood with NP-dimensionality. 2011. Vol. 57. In IEEE Transactions on Information Theory. pp.5467-5484.
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