Nonconcave penalized inverse regression in single-index models with high dimensional predictors
AbstractIn this paper we aim to estimate the direction in general single-index models and to select important variables simultaneously when a diverging number of predictors are involved in regressions. Towards this end, we propose the nonconcave penalized inverse regression method. Specifically, the resulting estimation with the SCAD penalty enjoys an oracle property in semi-parametric models even when the dimension, pn, of predictors goes to infinity. Under regularity conditions we also achieve the asymptotic normality when the dimension of predictor vector goes to infinity at the rate of pn=o(n1/3) where n is sample size, which enables us to construct confidence interval/region for the estimated index. The asymptotic results are augmented by simulations, and illustrated by analysis of an air pollution dataset.
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Bibliographic InfoArticle provided by Elsevier in its journal Journal of Multivariate Analysis.
Volume (Year): 100 (2009)
Issue (Month): 5 (May)
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Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description
Find related papers by JEL classification:
- 62H - - - - - -
- 62G - - - - - -
- Dim - Microeconomics - - - - -
- red - - - - - -
- Div - Microeconomics - - - - -
- par - - - - - -
- Inv - Health, Education, and Welfare - - - - -
- reg - - - - - -
- SCA - - - - - -
- Spa - - - - - -
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- Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
- Lexin Li, 2007. "Sparse sufficient dimension reduction," Biometrika, Biometrika Trust, vol. 94(3), pages 603-613.
- Liqiang Ni & R. Dennis Cook & Chih-Ling Tsai, 2005. "A note on shrinkage sliced inverse regression," Biometrika, Biometrika Trust, vol. 92(1), pages 242-247, March.
- Li, Bing & Wang, Shaoli, 2007. "On Directional Regression for Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 997-1008, September.
- Cook, R. Dennis & Ni, Liqiang, 2005. "Sufficient Dimension Reduction via Inverse Regression: A Minimum Discrepancy Approach," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 410-428, June.
- Lian, Heng & Li, Jianbo & Tang, Xingyu, 2014. "SCAD-penalized regression in additive partially linear proportional hazards models with an ultra-high-dimensional linear part," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 50-64.
- Hu, Yuao & Lian, Heng, 2013. "Variable selection in a partially linear proportional hazards model with a diverging dimensionality," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 61-69.
- Li, Gao-Rong & Zhu, Li-Ping & Zhu, Li-Xing, 2010. "Adaptive confidence region for the direction in semiparametric regressions," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1364-1377, July.
- Lai, Peng & Wang, Qihua & Zhou, Xiao-Hua, 2014. "Variable selection and semiparametric efficient estimation for the heteroscedastic partially linear single-index model," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 241-256.
- Li-Ping Zhu & Lin-Yi Qian & Jin-Guan Lin, 2011. "Variable selection in a class of single-index models," Annals of the Institute of Statistical Mathematics, Springer, vol. 63(6), pages 1277-1293, December.
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