  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.
