Simultaneous estimations of optimal directions and optimal transformations for functional data

Sliced inverse regression (SIR) is a powerful method to deal with a dimension reduction model. As well known, SIR is equivalent to a transformation-based projection pursuit, where the optimal directions are just the directions in SIR. In this paper, we consider simultaneous estimations of optimal directions for functional data and optimal transformations. We take a reproducing kernel Hilbert space approach. Both the directions and the transformations are chosen from reproducing kernel Hilbert spaces. A learning rate is established for the estimators.