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 Hölder 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  and  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)).
Bibliographical noteErratum: Optimal two-stage procedures for estimating location and size of the maximum of a multivariate regression function. Annals of Statistics (2012) 40 (2850-2876) DOI: 10.1214/12-AOS1053.
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