Uniform estimation of a signal based on inhomogeneous data

We want to reconstruct a signal based on inhomogeneous data (the amount of data can vary strongly), using the model of regression with a random design. Our aim is to understand the consequences of inhomogeneity on the accuracy of estimation within the minimax framework. Using the uniform metric weighted by a spatially-dependent rate as a benchmark for an estimator accuracy, we are able to capture the deformation of the usual minimax rate in situations with local lacks of data (modelled by a design density with vanishing points). In particular, we construct an estimator both design and smoothness adaptive, and a new criterion is developed to prove the optimality of these deformed rates.