Best estimation of functional linear models

Observations that are realizations of some continuous process are frequently found in science, engineering, economics, and other fields. In this paper, we consider linear models with possible random effects and where the responses are random functions in a suitable Sobolev space. In particular, the processes cannot be observed directly. By using smoothing procedures on the original data, both the response curves and their derivatives can be reconstructed, both as an ensemble and separately. From these reconstructed functions, one representative sample is obtained to estimate the vector of functional parameters. A simulation study shows the benefits of this approach over the common method of using information either on curves or derivatives. The main theoretical result is a strong functional version of the Gauss–Markov theorem. This ensures that the proposed functional estimator is more efficient than the best linear unbiased estimator (BLUE) based only on curves or derivatives.