2013/02/19 by Eduard Belitser, Subhashis Ghosal, Harry van Zanten
Mathematics · #math.ST #stat.TH
paper · pdf · doi:10.1214/12-aos1053
published as Annals of Statistics 2012, Vol. 40, No. 6, 2850-2876 · Published in at http://dx.doi.org/10.1214/12-AOS1053 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2013/02/19 · arxiv updated 2013/02/20
We propose a two-stage procedure for estimating the location \boldsμ and size M of the maximum of a smooth d-variate regression function f. In the first stage, a preliminary estimator of \boldsμ 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 \boldsμ and M. More specifically, for α-smooth regression functions, the optimal nonparametric rates n-(α-1)/(2α+d) and n-α/(2α+d) at the first stage can be improved to n-(α-1)/(2α) and n-1/2, respectively, for α>1+√(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 α>2 starting with a preliminary estimator with any power-law rate at the first stage.