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Functional sliced inverse regression (FSIR) is one of the most popularalgorithms for functional sufficient dimension reduction (FSDR). However, thechoice of slice scheme in FSIR is critical but challenging. In this paper, wepropose a new method called functional slicing-free inverse regression (FSFIR)to estimate the central subspace in FSDR. FSFIR is based on the martingaledifference divergence operator, which is a novel metric introduced tocharacterize the conditional mean independence of a functional predictor on amultivariate response. We also provide a specific convergence rate for theFSFIR estimator. Compared with existing functional sliced inverse regressionmethods, FSFIR does not require the selection of a slice number. Simulationsdemonstrate the efficiency and convenience of FSFIR.

Functional Slicing-free Inverse Regression via Martingale Difference Divergence Operator