Asymptotically minimax estimation of a function with jumps

Abstract

Asymptotically minimax nonparametric estimation of a regression function observed in white Gaussian noise over a bounded interval is considered, with respect to a L 2 -loss function. The unknown function f is assumed to be m times dierentiable except for an unknown, though nite, number of jumps, with piecewise mth derivative bounded in L 2 -norm. An estimator is constructed, attaining the same optimal risk bound, known as Pinsker's constant, as in the case of smooth functions (without jumps).