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Asymtotics of Dantzig selector for a general single-index model

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Abstract

As two popularly used variable selection methods, the Dantzig selector and the LASSO have been proved asymptotically equivalent in some scenarios. However, it is not the case in general for linear models, as disclosed in Gai, Zhu and Lin’s paper in 2013. In this paper, it is further shown that generally the asymptotic equivalence is not true either for a general single-index model with random design of predictors. To achieve this goal, the authors systematically investigate necessary and sufficient conditions for the consistent model selection of the Dantzig selector. An adaptive Dantzig selector is also recommended for the cases where those conditions are not satisfied. Also, different from existing methods for linear models, no distributional assumption on error term is needed with a trade-off that more stringent condition on the predictor vector is assumed. A small scale simulation is conducted to examine the performances of the Dantzig selector and the adaptive Dantzig selector.

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Correspondence to Yan Fan.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 11501354, 11201499, 11301309 and 71473280, and 2015 Shanghai Young Faculty Training Program under Grant No. A1A-6119-15-003.

This paper was recommended for publication by Editor SUN Liuquan.

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Fan, Y., Gai, Y. & Zhu, L. Asymtotics of Dantzig selector for a general single-index model. J Syst Sci Complex 29, 1123–1144 (2016). https://doi.org/10.1007/s11424-015-4124-y

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  • DOI: https://doi.org/10.1007/s11424-015-4124-y

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