Abstract
In this paper, we consider the adaptive group Lasso in high-dimensional linear regression. Some extensions have been done with other fitting procedures, such as adaptive Lasso, nonconcave penalized likelihood and adaptive elastic-net. Under appropriate conditions, we establish the consistency and asymptotic normality, which means that the adaptive group Lasso shares the oracle property in high-dimensional linear regression when the number of group variables diverges with the sample size.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Antoniadis A, Gijbels I, Lambert-Lacroix S (2014) Penalized estimation in additive varying coefficient models using grouped regularization. Stat Pap 55:727–750
Bach F (2008) Consistency of the group Lasso and multiple kernel learning. J Mach Learn Res 9:1179–1225
Chesneau C, Hebiri M (2008) Some theoretical results on the grouped variables Lasso. Math Methods Stat 17:317–326
Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96:1348–1360
Fan J, Peng H (2004) Nonconcave penalized likelihood with a diverging number of parameters. Ann Stat 32:928–961
Huang J, Horowitz JL, Ma S (2008) Asymptotic properties of bridge estimators in sparse high-dimensional regression models. Ann Stat 36:587–613
Huang J, Ma S, Zhang C-H (2008) Adaptive Lasso for sparse high-dimensional regression models. Stat Sin 18:1603–1618
Huber PJ (1973) Robust regression: asymptotics, conjectures and Monte Carlo. Ann Stat 1:799–821
Nardi Y, Rinaldo A (2008) On the asymptotic properties of the group Lasso estimator for linear models. Electron J Stat 2:605–633
Tibshirani R (1996) Regression shrinkage and selection via the Lasso. J R Stat Soc 58:267–288
Wang H, Leng C (2008) A note on adaptive group Lasso. Comput Stat Data Anal 52:5277–5286
Wang L, You Y, Lian H (July 2014) Convergence and sparsity of Lasso and group Lasso in high-dimensional generalized linear models. Stat Pap Published online
Wei F (2012) Group selection in high-dimensional partially linear additive model. Braz J Probab Stat 26:219–243
Wei F, Huang J (2010) Consistent group selection in high-diemnsional linear regression. Bernoulli 16:1369–1384
Yuan M, Lin Y (2006) Model selection and estimation in regression with grouped variables. J R Stat Soc 68:49–67
Zhao P, Yu B (2006) On model selection consistency of Lasso. J Mach Learn Res 7:2541–2567
Zou H (2006) The adaptive Lasso and its oracle properties. J Am Stat Assoc 101:1418–1429
Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc 67:301–320
Zou H, Zhang H (2009) On the adaptive elastic-net with a diverging number of parameters. Ann Stat 37:1733–1751
Acknowledgments
This project is supported in part by Natural Science Foundation of Zhejiang Province, China (LY14A010003) and National Natural Science Foundation of China (11101362). The authors are thankful to two anonymous referees for their constructive comments and useful suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, C., Xiang, Y. On the oracle property of adaptive group Lasso in high-dimensional linear models. Stat Papers 57, 249–265 (2016). https://doi.org/10.1007/s00362-015-0684-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-015-0684-0