Open Access
December 2012 Optimal two-stage procedures for estimating location and size of the maximum of a multivariate regression function
Eduard Belitser, Subhashis Ghosal, Harry van Zanten
Ann. Statist. 40(6): 2850-2876 (December 2012). DOI: 10.1214/12-AOS1053


We propose a two-stage procedure for estimating the location $\mathbf{\mu}$ and size $M$ of the maximum of a smooth $d$-variate regression function $f$. In the first stage, a preliminary estimator of $\mathbf{\mu}$ obtained from a standard nonparametric smoothing method is used. At the second stage, we “zoom-in” near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity. We fit an appropriate polynomial regression model to estimate the location and size of the maximum. We establish that, under suitable smoothness conditions and appropriate choice of the zooming, the second stage estimators have better convergence rates than the corresponding first stage estimators of $\mathbf{\mu}$ and $M$. More specifically, for $\alpha$-smooth regression functions, the optimal nonparametric rates $n^{-(\alpha-1)/(2\alpha+d)}$ and $n^{-\alpha/(2\alpha+d)}$ at the first stage can be improved to $n^{-(\alpha-1)/(2\alpha)}$ and $n^{-1/2}$, respectively, for $\alpha>1+\sqrt{1+d/2}$. These rates are optimal in the class of all possible sequential estimators. Interestingly, the two-stage procedure resolves “the curse of the dimensionality” problem to some extent, as the dimension $d$ does not control the second stage convergence rates, provided that the function class is sufficiently smooth. We consider a multi-stage generalization of our procedure that attains the optimal rate for any smoothness level $\alpha>2$ starting with a preliminary estimator with any power-law rate at the first stage.


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Eduard Belitser. Subhashis Ghosal. Harry van Zanten. "Optimal two-stage procedures for estimating location and size of the maximum of a multivariate regression function." Ann. Statist. 40 (6) 2850 - 2876, December 2012.


Published: December 2012
First available in Project Euclid: 8 February 2013

zbMATH: 1296.62158
MathSciNet: MR3097962
Digital Object Identifier: 10.1214/12-AOS1053

Primary: 62L12
Secondary: 62G05 , 62H12 , 62L05

Keywords: adaptive estimation , multi-stage procedure , optimal rate , sequential design , two-stage procedure

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • December 2012
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