Efficient estimation for time-dynamic longitudinal single-index model

We study the efficient estimation procedure of a new single-index model which can reflect the time-dynamic effects for longitudinal covariates. We propose a efficient estimator of the single-index parameter by using a feasible bias-corrected generalized estimating equation. In order to achieve this goal, we use the working independence estimator as an initial estimation, and then a non parametric smoothing technique is used to model the covariance matrix. With appropriate initial estimates of the parametric index, the proposed estimators are shown to be n$\sqrt{n}$-consistent and asymptotically normally distributed, and the two-stage estimator is shown to be more efficient than the first-stage estimator. We also address the non parametric estimation of regression functions and provide estimates with optimal convergence rates. The finite-sample properties of the estimator are illustrated by some simulation examples, as well as a real data application.