Oracle-efficient M-estimation for single-index models with a smooth simultaneous confidence band

Single-index models are important and popular semiparametric models, as they can handle the problem of the “curse of dimensionality” and enjoy the flexibility of nonparametric modeling and the interpretability of parametric modeling. Most existing methods for single-index models are sensitive to outliers or heavy-tailed distributions because they use the least squares criterion. An oracle-efficient M-estimator is proposed for single-index models, and a smooth simultaneous confidence band is constructed by treating the index coefficients as nuisance parameters. Under general assumptions it is shown that the M-estimator for the nonparametric link function, based on any \(\sqrt{n}\)-consistent coefficient index parameter estimators, is oracle-efficient. This means that it is uniformly as efficient as the infeasible one obtained by M-regression using the true single-index coefficient parameters. As a result, the asymptotic distribution of the maximal deviation between the M-type kernel estimator and the true link function is derived, and an asymptotically accurate simultaneous confidence band is established as a global inference tool for the link function. The proposed method generalizes the desirable uniform convergence property of ordinary least squares to the M-estimation. Meanwhile, it is a general approach that allows any \(\sqrt{n}\)-consistent coefficient parameter estimators to be applied in the procedure to make global inferences for the link function. Simulation studies with commonly encountered sample sizes are reported to support the theoretical findings. These numerical results show certain desirable robustness properties against heavy-tailed errors and outliers. As an illustration, the proposed method is applied to the analysis of a car purchasing dataset.