## 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 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 Y_{k} = f(x_{k})+ ξ_{k}, x_{k} ∈ D ⊂ R^{d}, k = 1,...,n, the assumptions and notation (like, ≲, etc.) from Belitser et al. (2012). Consider this model now under a fixed equidistant design x_{k} ∈ D ⊂ R^{d}, k = 1,...,n. Define R(H_{d}(α,D)) = inf _{M f}∈_{Hd}(α,D) sup E_{f}|M − M_{f}|, M_{f} = sup f(x), x∈D the infimum is taken over all estimators. The results of [1-3] suggest that (1) R(H_{d}(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(H_{d}(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 x_{0} ∈ D, the so-called pointwise estimation problem. The pointwise minimax rate is known to be R_{pw(}H_{d}(α,D)) = inf sup E_{f} ||f( x_{0})− f(x_{0})|| n^{−}α/(2α+d). f( x_{0})f∈H_{d}(α,D)learly, the former problem is not easier than the latter, which implies the first relation in (2): n^{−}α/(2α+^{d)} ≲ R_{pw}(H_{d}(α,D))≲ R(H_{d}(α,D)).

Original language | English |
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Pages (from-to) | 612-613 |

Number of pages | 2 |

Journal | Annals of Statistics |

Volume | 49 |

Issue number | 1 |

Early online date | 29 Jan 2021 |

DOIs | |

Publication status | Published - Feb 2021 |

### Bibliographical note

Erratum: 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.Publisher Copyright:

© 2021 Institute of Mathematical Statistics. All rights reserved.

Copyright:

Copyright 2021 Elsevier B.V., All rights reserved.