Abstract: In this paper, we propose simultaneous confidence bands for the nonparametric link function in single-index models in the presence of a nuisance index parameter. We establish the asymptotic properties for the link function and its derivative that allow simultaneous confidence bands for various inference tasks. In addition, we propose an adaptive Neyman test statistic for testing the linearity of the link function. We then conduct simulation studies to evaluate the performance of the proposed method, and apply them to two data sets for illustration.

Key words and phrases: Adaptive Neyman test, difference-based estimator, local linear smoother, residual variance, simultaneous confidence band, single-index model.