Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data

This paper considers nonparametric estimation in a varying coefficient mode l with repeated measurements (Y-ij,X-ij,t(ij)), for i=1,...,n and j=1,...,n i, where x(ij)=(X-ij0,..., X-ijk)(T) and (Y-ij,X-ij, t(ij)) denote the jth outcome, covariate and time design points, respectively, of the ith subject . The model considered here is Y-ij =X(ij)(T)beta(t(ij))+ epsilon(i)(t(ij)) , where beta(t) = (beta(0)(t),..., beta(k)(t))(T), for k greater than or eq ual to 0, are smooth nonparametric functions of interest and epsilon(i)(t) is a zero-mean stochastic process. The measurements are assumed to be indep endent for different subjects but can be correlated at different time point s within each subject. Two nonparametric estimators of beta(t), namely a sm oothing spline and a locally weighted polynomial, are derived for such repe atedly measured data. A crossvalidation criterion is proposed for the selec tion of the corresponding smoothing parameters. Asymptotic properties, such as consistency, rates of convergence and asymptotic mean squared errors, a re established for kernel estimators, a special case of the local polynomia ls. These asymptotic results give useful insights into the reliability of o ur general estimation methods. An example of predicting the growth of child ren born to HIV infected mothers based on gender, HIV status and maternal v itamin A levels shows that this model and the corresponding nonparametric e stimators are useful in epidemiological studies.