On global and pointwise adaptive estimation

Let an estimated function belong to a Lipschitz class of order a. 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 function can 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.