Abstract: Sparse functional data analysis (FDA) is powerful for making inference on the underlying random function when noisy observations are collected at sparse time points. To have a precise inference, knowledge on optimal designs that allow the experimenters to collect informative functional data is crucial. Here, we propose a framework for selecting optimal designs to precisely predict functional principal and empirical component scores. Our work gives a relevant generalization of previous results on the design for predicting individual response curves. We obtain optimal designs, and evaluate the performance of commonly used designs. We demonstrate that without a judiciously selected design, there can be a great loss in statistical efficiency.

Key words and phrases: Design efficiency, exact designs, mixed model equations, random function, sparse orthonormal approximation.