  We consider penalized extremum estimation of a high-dimensional, possibly
nonlinear model that is sparse in the sense that most of its parameters are
zero but some are not. We use the SCAD penalty function, which provides model
selection consistent and oracle efficient estimates under suitable conditions.
However, asymptotic approximations based on the oracle model can be inaccurate
with the sample sizes found in many applications. This paper gives conditions
under which the bootstrap, based on estimates obtained through SCAD
penalization with thresholding, provides asymptotic refinements of size \(O
\left( n^{- 2} \right)\) for the error in the rejection (coverage) probability
of a symmetric hypothesis test (confidence interval) and \(O \left( n^{- 1}
\right)\) for the error in the rejection (coverage) probability of a one-sided
or equal tailed test (confidence interval). The results of Monte Carlo
experiments show that the bootstrap can provide large reductions in errors in
rejection and coverage probabilities. The bootstrap is consistent, though it
does not necessarily provide asymptotic refinements, even if some parameters
are close but not equal to zero. Random-coefficients logit and probit models
and nonlinear moment models are examples of models to which the procedure
applies.
