OPTIMAL TWO-STAGE PROCEDURES FOR ESTIMATING LOCATION AND SIZE OF THE MAXIMUM OF A MULTIVARIATE REGRESSION FUNCTION

• 成果类型: Article
• 署名作者: Belitser, Eduard; Ghosal, Subhashis; van Zanten, Harry
• 署名单位: Eindhoven University of Technology; North Carolina State University; University of Amsterdam
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1053
• 发表日期: 2012
• 页码: 2850-2876
• 关键词: mode

摘要

We propose a two-stage procedure for estimating the location mu and size M of the maximum of a smooth d-variate regression function f. In the first stage, a preliminary estimator of 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 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 + root 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.