an:05266814
Zbl 1133.62048
Huang, Jian; Horowitz, Joel L.; Ma, Shuangge
Asymptotic properties of bridge estimators in sparse high-dimensional regression models
EN
Ann. Stat. 36, No. 2, 587-613 (2008).
00218196
2008
j
62J05 62F12 62E20 62J07 62H12
penalized regression; high-dimensional data; variable selection; asymptotic normality; oracle property
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 \textit{J. Fan} and \textit{R. Li} [J. Am. Stat. Assoc. 96, No. 456, 1348--1360 (2001; Zbl 1073.62547)] and \textit{J. Fan} and \textit{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.
Zbl 1073.62547; Zbl 1092.62031