In semivarying coefficient modeling of longitudinal/clustered data, of primary interest is usually the parametric component which involves unknown constant coefficients. First, we study semiparametric efficiency bound for estimation of the constant coefficients in a general setup. It can be achieved by spline regression using the true within-subject covariance matrices, which are often unavailable in reality. Thus, we propose an estimator when the covariance matrices are unknown and depend only on the index variable. First, we estimate the covariance matrices using residuals obtained from a preliminary estimation based on working independence and both spline and local linear regression. Then, using the covariance matrix estimates, we employ spline regression again to obtain our final estimator. It achieves the semiparametric efficiency bound under normality assumption and has the smallest asymptotic covariance matrix among a class of estimators even when normality is violated. Our theoretical results hold either when the number of within-subject observations diverges or when it is uniformly bounded. In addition, using the local linear estimator of the nonparametric component is superior to using the spline estimator in terms of numerical performance. The proposed method is compared with the working independence estimator and some existing method via simulations and application to a real data example.
"Efficient estimation in semivarying coefficient models for longitudinal/clustered data." Ann. Statist. 44 (5) 1988 - 2017, October 2016. https://doi.org/10.1214/15-AOS1385