Efficient estimation in heteroscedastic single-index models

In this article, we focus on the efficient estimation in single-index models with heteroscedastic errors. We first develop a nonparametric estimator of the variance function based on a fully nonparametric function or a dimension reduction structure, and the resulting estimator is consistent. Then, we propose a reweighting estimator of the parametric component via taking the estimated variance function into account, and the main results show that it has a smaller asymptotic variance than the naive estimator that neglects the heteroscedasticity. Simulation studies are conducted to evaluate the efficacy of the proposed methodologies, and an analysis of a real data example is provided for illustration.