Two-Step Estimation of Functional Linear Models with Applications to Longitudinal Data

Functional linear models are useful in longitudinal data analysis. They include many classical and recently proposed statistical models for longitudinal data and other functional data. Recently, smoothing spline and kernel methods have been proposed for estimating their coe cient functions nonparametrically but these methods are either intensive in computation or ine cient in performance. To overcome these drawbacks, in this paper, a simple and powerful two-step alternative is proposed. In particular, the implementation of the proposed approach via local polynomial smoothing is discussed. Methods for estimating standard deviations of estimated coe cient functions are also proposed. Some asymptotic results for the local polynomial estimators are established. Two longitudinal data sets, one of which involves time-dependent covariates, are used to demonstrate the proposed approach. Simulation studies show that our two-step approach improves the kernel method proposed in Hoover, et al (1998) in several aspects such as accuracy, computation time and visual appealingness of the estimators.