Sharp estimation in sup norm with random design

In this paper, we study the estimation of a function based on noisy inhomogeneous data (the amount of data can vary on the estimation domain). We consider the model of regression with random design, where the design density is unknown. We construct an asymptotically sharp estimator which converges, for sup norm error loss, with a spatially dependent normalisation which is sensitive to the variations in the local amount of data. This estimator combines both kernel and local polynomial methods, and it does not depend within its construction on the design density. Then, we prove that the normalisation is optimal in an appropriate sense.