Asymptotic properties of bridge estimators in sparse high-dimensional regression models

TL;DR

This paper investigates the asymptotic behavior of bridge estimators in high-dimensional linear regression, demonstrating their ability to correctly identify relevant covariates and achieve oracle properties under certain conditions.

Contribution

It establishes conditions under which bridge estimators possess oracle properties in high-dimensional settings, including when the number of covariates exceeds the sample size under partial orthogonality.

Findings

Bridge estimators correctly select nonzero coefficients with high probability.

Establishes oracle properties for bridge estimators in high-dimensional models.

Partial orthogonality allows consistent variable selection even when covariates outnumber samples.

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.