Nonparametric regression and spatially inhomogeneous information.

Summary We study the nonparametric estimation of a signal based on inhomogeneous noisy data (the amount of data varies on the estimation domain). We consider the model of nonparametric regression with random design. Our aim is to understand the consequences of inhomogeneous data on the estimation problem in the minimax setup. Our approach is twofold: local and global. In the local setup, we want to recover the regression at a point with little, or much data. By translating this property into several assumptions on the design density, we obtain a large range of new minimax rates, containing very slow and very fast rates. Then, we construct a smoothness adaptive procedure, and we show that it converges with a minimax rate penalised by a minimal cost. In the global setup, we want to recover the regression with sup norm loss. We propose estimators converging with rates which are sensitive to the inhomogeneous behaviour of the information in the model. We prove the spatial optimality of these rates, which consists in an enforcement of the classical minimax lower bound for sup norm loss. In particular, we construct an asymptotically sharp estimator over Hölder balls with any smoothness, and a confidence band with a width which adapts to the local amount of data.