Nonparametric Covariance Function Estimation for Functional and Longitudinal Data

Covariance function plays a critical role in functional and longitudinal data analysis. In this paper, we consider nonparametric covariance function estimation using a reproducing kernel Hilbert space framework. A regularization method is introduced through a careful characterization of the function space in which a covariance function resides. It is shown that the procedure enjoys desirable theoretical and numerical properties. In particular, even though the covariance function is bivariate, the rates of convergence attained by the regularization method are very similar to those typically achieved for estimating univariate functions. Our results generalize and improve some of the known results in the literature both for estimating the covariance function and for estimating the functional principal components. The procedure is easy to implement and its numerical performance is investigated using both simulated and real data. In particular our method is illustrated in an analysis of a longitudinal CD4 count data from an HIV study.