Translator Disclaimer
April 2008 Asymptotic properties of bridge estimators in sparse high-dimensional regression models
Jian Huang, Joel L. Horowitz, Shuangge Ma
Ann. Statist. 36(2): 587-613 (April 2008). DOI: 10.1214/009053607000000875

Abstract

We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero. We show that under appropriate conditions, bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance. Thus, bridge estimators have an oracle property in the sense of Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348–1360] and Fan and Peng [Ann. Statist. 32 (2004) 928–961]. In general, the oracle property holds only if the number of covariates is smaller than the sample size. However, under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients, we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size.

Citation

Download Citation

Jian Huang. Joel L. Horowitz. Shuangge Ma. "Asymptotic properties of bridge estimators in sparse high-dimensional regression models." Ann. Statist. 36 (2) 587 - 613, April 2008. https://doi.org/10.1214/009053607000000875

Information

Published: April 2008
First available in Project Euclid: 13 March 2008

zbMATH: 1133.62048
MathSciNet: MR2396808
Digital Object Identifier: 10.1214/009053607000000875

Subjects:
Primary: 62J05, 62J07
Secondary: 60F05, 62E20

Rights: Copyright © 2008 Institute of Mathematical Statistics

JOURNAL ARTICLE
27 PAGES


SHARE
Vol.36 • No. 2 • April 2008
Back to Top