FUNCTIONAL SLICED INVERSE REGRESSION IN A REPRODUCING KERNEL HILBERT SPACE: A THEORETICAL CONNECTION TO FUNCTIONAL LINEAR REGRESSION

We consider functional sliced inverse regression (FSIR) when the functional indices are assumed to be elements of a reproducing kernel Hilbert space (RKHS). This work is motivated by a prior study on functional linear regression (FLR) that incorporates a penalty involving the RKHS norm. Utilizing a close connection between FLR and FSIR not noted before, we show that the FSIR can be dealt with by an analogy with the FLR. Methodologically, this is straightforward, but the corresponding theoretical transfer from the FLR to the FSIR is nontrivial. In particular, we show that the convergence rate for the FSIR is the same as that of the FLR, and is thus minimax. This result is particularly interesting given the far more general specification of dimension-reduction problems compared with that of FLR. Simulations and real data are used to compare this with the functional PCA-based approach, where the functional index is expanded using the eigenfunctions of the covariance kernel.