Doubly robust and efficient estimation in a partially linear multiple-index model

In this paper, we introduce a partially linear multiple-index model. A new loss function is proposed to set up a general class of estimating equations to obtain the doubly robust and efficient estimation of the regression parameters. The estimators of the nonparametric functions are also constructed via local linear fitting. The asymptotic properties of the proposed estimators are proved. Furthermore, the bias-corrected empirical log-likelihood ratios of the regression parameters are proposed. It is shown that the proposed ratios are asymptotically standard chi-squared, and the obtained results can be directly used to construct the confidence regions of the regression parameters. Simulation studies and real data analysis show that the proposed method is effective.