Efficient Estimation in Partially Linear Single-Index Models for Longitudinal Data

In this paper, we consider the estimation of both the parameters and the nonparametric link function in partially linear single-index models for longitudinal data which may be unbalanced. In particular, a new three-stage approach is proposed to estimate the nonparametric link function using marginal kernel regression and the parametric components with generalized estimating equations. The resulting estimators properly account for the within-subject correlation. We show that the parameter estimators are asymptotically semiparametric efficient. We also show that the asymptotic variance of the link function estimator is minimized when the working error covariance matrices are correctly specified. The new estimators are more efficient than estimators in the existing literature. These asymptotic results are obtained without assuming normality. The finite sample performance of the proposed method is demonstrated by simulation studies. In addition, two real data examples are analyzed to illustrate the methodology.