Abstract: The aim of this paper is to recover a signal based on inhomogeneous noisy data (the amount of data can vary strongly from one point to another.) In particular, we focus on the understanding of the consequences of the inhomogeneity of the data on the accuracy of estimation. For that purpose, we consider the model of regression with a random design, and we consider the minimax framework. Using the uniform metric weighted by a spatially-dependent rate in order to assess the accuracy of estimators, we are able to capture the deformation of the usual minimax rate in situations where local lacks of data occur (the latter are modelled by a design density with vanishing points). In particular, we construct an estimator both design and smoothness adaptive, and we develop a new criterion to prove the optimality of these deformed rates.
Key words and phrases: Adaptive estimation, minimax theory, nonparametric regression, random design.