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Convergence Rates of Density Estimation in Besov Spaces
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Applied Mathematics·Volume 02 (2011)·Pages 1258–1262·Published 11 October 2011·DOI10.4236/am.2011.210175
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Abstract
The optimality of a density estimation on Besov spaces <i>B<sup>s</sup><sub>r,q</sub>(R)</i> for the <i>L<sub>p</sub></i> risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence <i>R<sub>n</sub>(B<sup>s</sup><sub>r,q</sub>, p)</i>, they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.
KeywordsOptimal Rate of ConvergenceDensity EstimationBesov SpacesWavelets
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