Estimation of the Effective Dimension Reduction Subspace

Yi = f(xi) + εi = g(Θ >xi) + εi, i = 1, . . . , n, is addressed. In the general setup we are interested in, the covariates xi ∈ R, Θ is a d×m orthogonal matrix (ΘΘ = Im∗) and g : R ∗ → R is an unknown function. To be able to estimate Π consistently, we assume that S = Im(Θ) is the smallest subspace satisfying f(xi) = f(ΠSxi), ∀i = 1, . . . , n, where ΠS stands for the orthogonal projector in R onto the subspace S. We will focus our attention on the case where m is known. Many methods dealing with the estimation of the EDR subspace perform principal component analysis on a family of vectors, say β̂1, . . . , β̂L, nearly lying in the EDR subspace. This is in particular the case for the structure-adaptive approach proposed by Hristache, Juditsky, Polzehl and Spokoiny (Ann. Statist. 2001). In contrast with this approach, we propose to estimate the projector onto the EDR subspace by the solution to the optimization problem