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June 1998 On global and pointwise adaptive estimation
Sam Efromovich
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Bernoulli 4(2): 273-282 (June 1998).


Let an estimated function belong to a Lipschitz class of order α . 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 α is unknown, then the minimax mean squared (pointwise) error convergence slows down from n - 2α/(2α+1) for the case of the given α to [ n/ln(n)] - 2α/(2α+1) . At the same time, the minimax mean integrated squared (global) error convergence is proportional to n - 2α/(2α+1) for the cases of known and unknown α . 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.


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Sam Efromovich. "On global and pointwise adaptive estimation." Bernoulli 4 (2) 273 - 282, June 1998.


Published: June 1998
First available in Project Euclid: 26 March 2007

zbMATH: 0908.62046
MathSciNet: MR1632991

Keywords: analytic and Lipschitz functions , efficiency , mean integrated squared error , mean squared error

Rights: Copyright © 1998 Bernoulli Society for Mathematical Statistics and Probability

Vol.4 • No. 2 • June 1998
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