The asymptotic minimax constant for sup-norm loss in nonparametric density estimation

Abstract

We develop the exact constant of the risk asymptotics in the uniform norm for density estimation. This constant has already been found for nonparametric regression and for signal estimation in Gaussian white noise. Hölder classes for arbitrary smoothness index β>0 on the unit interval are considered. The constant involves the value of an optimal recovery problem as in the white noise case, but in addition it depends on the maximum of densities in the function class.