© 1998 by Biometrika Trust
Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data
Department of Epidemiology, The John Hopkins University Baltimore, Maryland 21205, U.S.A.dhoover{at}share.sph.jhu.edu
Department of Statistics, University of California Berkeley, California 94720, U.S.A.rice{at}stat.berkeley.edu
Department of Mathematical Sciences, The Johns Hopkins University Baltimore, Maryland 21218, U.S.A.colinwu{at}columbo.mts.jhu.edu
Department of Epidemiology, The Johns Hopkins University Baltimore, Maryland 21205, U.S.A.lpyang{at}share.sph.jhu.edu
This paper considers nonparametric estimation in a varying coefficient model with repeated measurements (Yij, Xij, tij), for i = 1,..., ni and j = 1,..., ni, where Xij=(Xijo,..., xijk)T and (Yij, Xij, tij) denote the jth outcome, covariate and time design points, respectively, of the ith subject. The model considered here is Yij = XijTß(tij) + +i(tij), where ß(t) = (ß0(t), ..., Bk(t))T, for k 
0, are smooth nonparametric functions of interest and 
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 ß(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.
Key Words: Longitudinal data • Mean squared error • Nonparametric estimation • Rates of convergence • Varying coefficient model
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