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

This paper considers nonparametric estimation in a varying coefficient model with repeated measurements (Y-ij,X-ij,t(ij)), for i=1,...,n and j=1,...,ni, 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 equal to 0, are smooth nonparametric functions of interest and epsilon(i)(t) is a zero-mean stochastic process. The measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject. Two nonparametric estimators of beta(t), namely a smoothing spline and a locally weighted polynomial, are derived for such repeatedly measured data. A crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters. Asymptotic properties, such as consistency, rates of convergence and asymptotic mean squared errors, are established for kernel estimators, a special case of the local polynomials. These asymptotic results give useful insights into the reliability of our general estimation methods. An example of predicting the growth of children born to HIV infected mothers based on gender, HIV status and maternal vitamin A levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies.