Abstract: Statistical inference for high-dimensional regression models is a challenging problem. Existing methods focus on inference for finite-dimensional components of the model parameters. Constructing the parameter estimators and establishing the asymptotic inference are specific to each model. In this study, we treat a high-dimensional model as a special case of a semiparametric model. We propose a general framework for constructing one-step regularized estimators for any smooth functional of high-dimensional parameters, which can be viewed as an extension of the one-step efficient estimator for semiparametric models to an M-estimation in the high-dimensional model setting. We show that the proposed estimator is asymptotically normal under some general regularity conditions. We apply the proposed method to an inference for the coefficients in a high-dimensional lasso regression, and to determine the l²-norm of the functional coefficients in a high-dimensional additive model, allowing the number of covariates to grow exponentially with the sample size. A simulation study and a microarray data example are presented to demonstrate the performance of the proposed method.

Key words and phrases: Confidence intervals, high-dimension regression, M-estimation, one-step regularized estimators, semiparametric model.