Estimation and Inference of Semiparametric Single-Index Models with High-Dimensional Covariates

This paper develops new estimation and inference methods of high-dimensional single-index models. We propose a simple two-stage estimation method based on the average derivative estimator (ADE). This ADE is composed of weighted score functions of covariates that can easily be estimated under a semiparametric Gaussian copula structure. In the first stage, we plug in standard nonparametric estimates for marginal features and a regularized estimator for the precision matrix of the Gaussian copula to obtain high-dimensional score functions. In the second stage, we conduct LASSO-type thresholding to get sparse estimates of the regression coefficients in single-index models. Both stages involve only convex minimization problems. We derive the non-asymptotic bound of our estimator. For inference, we prove the asymptotic normality of a de-biased estimator using the one-step Newton-Raphson update. Our inferential tools do not rely on the Gaussian copula restriction and are more generally applicable with other pilot estimators.