Adaptively Local One-Dimensional Subproblems with Application to a Deconvolution Problem

Abstract

In this paper, a method for finding global minimax lower bounds is introduced. The idea is to adjust automatically the direction of a local one-dimensional subproblem at each location to the nearly hardest one, and to use locally the difficulty of the one-dimensional subproblem. This method has the advantages of being easily implemented and understood. The lower bound is then applied to nonparametric deconvolution to obtain the optimal rates of convergence for estimating a whole function. Other applications are also addressed.