Sharp estimation in sup norm with random design

The aim of this paper is to recover the regression function with sup norm loss. We construct an asymptotically sharp estimator which converges with the spatially dependent rate r_{n, \mu}(x) = P \big(\log n / (n \mu(x)) \big)^{s / (2s + 1)}, where $\mu$ is the design density, $s$ the regression smoothness, $n$ the sample size and $P$ is a constant expressed in terms of a solution to a problem of optimal recovery as in Donoho (1994). We prove this result under the assumption that $\mu$ is positive and continuous. This estimator combines kernel and local polynomial methods, where the kernel is given by optimal recovery, which allows to prove the result up to the constants for any $s>0$. Moreover, the estimator does not depend on $\mu$. We prove that $r_{n, \mu}(x)$ is optimal in a sense which is stronger than the classical minimax lower bound. Then, an inhomogeneous confidence band is proposed. This band has a non constant length which depends on the local amount of data.