Statistica Sinica 10(2000), 51-71

NONPARAMETRIC FUNCTION ESTIMATION OF THE

RELATIONSHIP BETWEEN TWO REPEATEDLY

MEASURED VARIABLES

A. F. Ruckstuhl$^*$, A. H. Welsh$^*$ and R. J. Carroll$^\dagger$

$^*$Australian National University and $^\dagger$Texas A& M University

Abstract: We describe methods for estimating the regression function nonparametrically, and for estimating the variance components in a simple variance component model which is sometimes used for repeated measures data or data with a simple clustered structure. We consider a number of different ways of estimating the regression function. The main results are that the simple pooled estimator which treats the data as independent performs very well asymptotically, but that we can construct estimators which perform better asymptotically in some circumstances. The local linear version of the quasi-likelihood estimator is supposed to exploit the covariance structure of the model but does not in fact do so, asymptotically performing worse than the simple pooled estimator.

Key words and phrases: Local linear regression, local quasi-likelihood estimator, semiparametric estimation, smoothing, variance components.