A note on smoothed functional inverse regression

Estimation in the context of functional data analysis is almost always non-parametric, since the object to be estimated lives in an infinite dimensional space. That is the case for the functional linear model with a real response and a process as covariables. In a recent paper Ferré and Yao state that the estimation of the Effective Dimension Reduction (EDR) subspace via SIR has parametric order. We show that a strong condition is needed for their statement to be true.