  This study proposes a novel method for estimation and hypothesis testing in
high-dimensional single-index models. We address a common scenario where the
sample size and the dimension of regression coefficients are large and
comparable. Unlike traditional approaches, which often overlook the estimation
of the unknown link function, we introduce a new method for link function
estimation. Leveraging the information from the estimated link function, we
propose more efficient estimators that are better aligned with the underlying
model. Furthermore, we rigorously establish the asymptotic normality of each
coordinate of the estimator. This provides a valid construction of confidence
intervals and $p$-values for any finite collection of coordinates. Numerical
experiments validate our theoretical results.
