Simultaneous confidence bands and global inferences for extended partially linear single-index models

Abstract

Substantial efforts have been made for the popular semiparametric single-index models in the last two decades. Extended partially linear single-index models as generalized single-index semiparametric models effectively reduce the problem of model misspecification by allowing the data to automatically choose the principal linear component and the principal nonlinear component. In this paper, we construct an oracle-efficient simultaneous confidence band as a global inference tool for the extended partially linear single-index models. Specifically, we apply a LASSO penalized local linear smoothing method to estimate the coefficient parameters and conduct variable selection to avoid model overfitting caused by the two appearances of covariates. It is shown that the local linear estimator for the nonparametric link function by employing the penalized coefficient estimates is oracle-efficient in the sense that it is uniformly as efficient as the ideal one obtained by utilizing the true coefficient parameters. Then by applying the oracle efficiency and the extreme value distribution theory of the local linear regression, an asymptotically accurate simultaneous confidence band for the nonparametric link function in the extended partially linear single-index models is established. Simulation experiments with commonly encountered sample sizes corroborate our theoretical findings. The proposed method is applied to analyze two engineering datasets: motor trend car road tests dataset and concrete slump tests dataset.