On global and pointwise adaptive estimation

Abstract

Let an estimated function belong to a Lipschitz class of order $\alpha$. Consider a minimax approach where the infimum is taken over all possible estimators and the supremum is taken over the considered class of estimated functions. It is known that, if the order $\alpha$ is unknown, then the minimax mean squared (pointwise) error convergence slows down from $n^{-2\alpha /(2\alpha +1)}$ for the case of the given $\alpha$ to $[n/ln(n){]}^{-2\alpha /(2\alpha +1)}$. At the same time, the minimax mean integrated squared (global) error convergence is proportional to $n^{-2\alpha /(2\alpha +1)}$ for the cases of known and unknown $\alpha$. We show that a similar phenomenon holds for analytic functions where the lack of knowledge of the maximal set to which the functioncan be analytically continued leads to the loss of a sharp constant. Surprisingly, for the more general adaptive minimax setting where we consider the union of a range of Lipschitz and a range of analytic functions neither pointwise error convergence nor global error convergence suffers an additional slowing down.