A sharp minimax lower bound for the nonparametric estimation of Sobolev densities of order 12

Results on sharp minimax estimation of the probability density from a Sobolev function class, established in the 1980s, were among the first sharp minimax results in the nonparametric curve estimation literature. This paper considers the last unsolved case — a Sobolev class of order 12 of densities with infinite support. This brings a closure to this classical topic in the nonparametric curve estimation theory.