Adaptive profile-empirical-likelihood inferences for generalized single-index models

We study generalized single-index models and propose an efficient equation for estimating the index parameter and unknown link function, deriving a quasi-likelihood-based maximum empirical likelihood estimator (QLMELE) of the index parameter. We then establish an efficient confidence region for any components of the index parameter using an adaptive empirical likelihood method. A pointwise confidence interval for the unknown link function is also established using the QLMELE. Compared with the normal approximation proposed by Cui et al. [Ann Stat. 39 (2011) 1658], our approach is more attractive not only theoretically but also empirically. Simulation studies demonstrate that the proposed method provides smaller confidence intervals than those based on the normal approximation method subject to the same coverage probabilities. Hence, the proposed empirical likelihood is preferable to the normal approximation method because of the complicated covariance estimation. An application to a real data set is also illustrated.