Functional sufficient dimension reduction: Convergence rates and multiple functional case

•Established the convergence rate for functional SIR and SAVE.•Consider multiple functional dimension reduction.•Numerical examples are constructed to show the performance of the estimators.

Although sufficient dimension reduction for functional data has received some attention in the literature, its theoretical properties are less understood. Besides, the current literature only focused on sliced inverse regression (SIR). In this paper we consider functional version of SIR and SAVE (sliced average variance estimation) via a Tikhonov regularization approach. Besides consistency, we show that their convergence rates are the same as the minimax rates for functional linear regression, which we think is an interesting theoretical result given that sufficient dimension reduction is much more flexible than functional linear regression. In sufficient dimension reduction, it is well known that estimation of multiple directions requires extraction of multiple eigenfunctions. We also consider multiple functional dimension reduction, in which one eigenfunction surprisingly recovers multiple index functions at once, despite its similarity with single functional case. The numerical properties are illustrated using several simulation examples as well as a Japanese weather dataset.