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aug 2006 Nonlinear estimation over weak Besov spaces and minimax Bayes method
Vincent Rivoirard
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Bernoulli 12(4): 609-632 (aug 2006). DOI: 10.3150/bj/1155735929

Abstract

Weak Besov spaces play an important role in statistics as maxisets of classical procedures or for measuring the sparsity of signals. The goal of this paper is to study weak Besov balls WB s ,p,q (C) from the statistical point of view by using the minimax Bayes method. In particular, we compare weak and strong Besov balls statistically. By building an optimal Bayes wavelet thresholding rule, we first establish that, under suitable conditions, the rate of convergence of the minimax risk for WB s ,p,q (C) is the same as for the strong Besov ball B s ,p,q (C) that is contained in WB s ,p,q (C) . However, we show that the asymptotically least favourable priors of WB s ,p,q (C) that are based on Pareto distributions cannot be asymptotically least favourable priors for B s ,p,q (C) . Finally, we present sample paths of such priors that provide representations of the worst functions to be estimated for classical procedures and we give an interpretation of the roles of the parameters s , p and q of WB s ,p,q (C) .

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Vincent Rivoirard. "Nonlinear estimation over weak Besov spaces and minimax Bayes method." Bernoulli 12 (4) 609 - 632, aug 2006. https://doi.org/10.3150/bj/1155735929

Information

Published: aug 2006
First available in Project Euclid: 16 August 2006

zbMATH: 1125.62001
MathSciNet: MR2248230
Digital Object Identifier: 10.3150/bj/1155735929

Keywords: asymptotically least favourable priors , Bayes method , minimax risk , rate of convergence , thresholding rules , weak Besov spaces

Rights: Copyright © 2006 Bernoulli Society for Mathematical Statistics and Probability

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Vol.12 • No. 4 • aug 2006
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