Abstract
Let an estimated function belong to a Lipschitz class of order . 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 is unknown, then the minimax mean squared (pointwise) error convergence slows down from for the case of the given to . At the same time, the minimax mean integrated squared (global) error convergence is proportional to for the cases of known and unknown . 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.
Citation
Sam Efromovich. "On global and pointwise adaptive estimation." Bernoulli 4 (2) 273 - 282, June 1998.
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