Correction note: “Optimal two-stage procedures for estimating location and size of the maximum of a multivariate regression function” Ann. Statist. 40 (2012) 2850–2876

Abstract

In the paper of Belitser, Ghosal and van Zanten (Ann. Statist. 40 (2012) 2850-2876), on page 2851 it is stated that “the minimax rate for estimating the maximum value of the function ranging over an α-smooth nonparametric class (e.g., isotropic Holder class defined below) is n−α/(2α+d)” (the rate n−α/(2α+d) is also mentioned in abstract, on pp. 2553 and 2859). This is not correct: instead of “n−α/(2α+d)” one should read “n−α/(2α+d) up to a log factor.” Recall the model Yk = f(xk)+ ξk, xk ∈ D ⊂ Rd, k = 1,...,n, the assumptions and notation (like, ≲, etc.) from Belitser et al. (2012). Consider this model now under a fixed equidistant design xk ∈ D ⊂ Rd, k = 1,...,n. Define R(Hd(α,D)) = inf M f∈Hd(α,D) sup Ef|M − Mf|, Mf = sup f(x), x∈D the infimum is taken over all estimators. The results of [1-3] suggest that (1) R(Hd(α,D)) (n/logn)−α/(2α+d). We call (1) conjecture for now, because, despite our extensive search in the literature, we were unable to find an exact reference that establishes (or implies) (1). The results [1] and [3] are obtained only for the one dimensional case in the white noise model, so formally they do not provide a conclusive proof of (1). Below we provide a proof for the upper bound. The lower bound should proceed in the same way as for the problem of estimating the function in the sup-norm. In essence, the difficulty of the problem of estimating the maximal values of a function is the same as that of the problem of estimating the function in the sup-norm. Here, we provide a short argument for the logarithmic sandwich bound n−α/(2α+d) ≲ R(Hd(α,D)) ≲ (n/logn)−α/(2α+d) (2). The first inequality of (2) can be argued by comparing the problem of estimating the maximal values of a function to the problem of estimating a function value at a fixed point x0 ∈ D, the so-called pointwise estimation problem. The pointwise minimax rate is known to be Rpw(Hd(α,D)) = inf sup Ef ||f( x0)− f(x0)|| n−α/(2α+d). f( x0)f∈Hd(α,D)learly, the former problem is not easier than the latter, which implies the first relation in (2): n−α/(2α+d) ≲ Rpw(Hd(α,D))≲ R(Hd(α,D)).