| In fixed-design kernel nonparametric regression, there has been a paucity of results for models which allow for correlated errors. Consider the following repeated-measurements model, applicable in growth curve analysis: Y(,s)(x(,t)) = g(x(,t)) + (epsilon)(,s)(x(,t)), s = 1,...,m (e.g., subjects), t = 1,...,n (e.g., time points) with errors of zero mean and within-subject covariance matrix (SIGMA). More specifically, we assume that cov (epsilon)(,s)(x(,t)),(epsilon)(,u)(x(,v)) = (delta)(,su)(sigma)(x(,t),x(,v)) where (delta)(,su) is the Kronecker-delta and (sigma)(x(,t),x(,v)) is the (t,v)('th) element of (SIGMA). Furthermore, it is assumed that (sigma)(x(,t),x(,v)) may be represented as the product of a scalar variance term and a suitably restricted correlation function (gamma)(x(,t) - x(,v)). Kernel estimators of the population regression function g(x) and its p('th) order derivatives are examined for specific as well as more general correlation functions. Limiting forms of an optimal linear combination of the subject means (and its measure of error) are derived. Necessary and sufficient conditions for consistency are stated for a general linear estimator for the Ornstein-Uhlenbeck correlation function, and sufficient conditions are given for a more geneal covariance structure. For the case p = 0, a numerical study investigating the requisite amount of smoothing and the efficiency of four popular kernel estimators is carried out. Asymptotic expansions of the mean squared error of the Gasser-Muller estimator of an arbitrary p('th) derivative are obtained for two general classes of correlation functions. Consistency and other results based on such expansions are discussed for orders p = 1 and p = 2. |
