Optimal Two-stage Procedures for Estimating Location and Size of Maximum of Multivariate Regression Functions

We propose a two-stage procedure for estimating the location μ and size M of the maximum of a smooth d-variate regression function f(x). Our procedure is motivated by Tang, Banerjee and Michailidis (2011). In the first stage, a preliminary estimator of μ 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 μ and M . More specifically, for α-smooth regression functions with α > 1 + √ 1 + d/2, 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, which are the optimal rates in the class of all possible sequential estimators. Interestingly, the two-stage procedure resolves the curse of 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. Adaptive properties of multi-stage approach are also discussed.