Asymptotic properties of bridge estimators in sparse high-dimensional regression models. (English) Zbl 1133.62048

Summary: 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 J. Fan and R. Li [J. Am. Stat. Assoc. 96, No. 456, 1348–1360 (2001; Zbl 1073.62547)] and J. Fan and H. Peng [Ann. Stat. 32, No. 3, 928–961 (2004; Zbl 1092.62031)]. 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.


62J05 Linear regression; mixed models
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
62J07 Ridge regression; shrinkage estimators (Lasso)
62H12 Estimation in multivariate analysis
Full Text: DOI arXiv


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