Data in the form of functions or curves are increasingly common in the sciences. We assume that per subject or unit one observes the realization of a square integrable stochastic process, either as a response or as a predictor. Several regression models for such infinite-dimensional data will be discussed. These include the functional linear model and functional least squares. Existence of solutions and their representation in functional canoncial components will be discussed.
For the case where the response is a function and the predictors are vectors, we consider a model based on the eigenfunction decomposition of the response function. Each principal component of the random response function is assumed to be a function of finite-dimensional predictors. Thus the problem is broken down into a series of classical regression problems with possibly high-dimensional predictors.
We aim to address these classical regression problems without having to specify a fully parametric model, while still escaping the curse of dimension. For this purpose we use a class of single index models which have been termed QLUEs for quasi-likelihood with nonparametric link and variance function estimation. The proposed methods are illustrated with data on reproduction and lifespan of medflies (Co-authors for the various parts include J. Carey, J. Chiou, G. He, and J.L. Wang).
| E-Mail: | mueller@wald.ucdavis.edu |