Simpler proofs for functional sliced inverse regression

In this short communication, we demonstrate that functional sliced inverse regression obtains the same convergence rate as that presented in Hall, P & Horowitz, JL (2007), ‘Methodology and convergence rates for functional linear regression’, Annals of Statistics, 35(1), 70–91 for functional linear regression, both based on functional principal component analysis. This result is interesting because functional sliced inverse regression imposes far fewer structural constraints than functional linear regression, including an unknown link function. The two proofs provided emphasize the similarity between the two, and the proofs are thus much simpler than typically found in the literature.