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April 2006 Adaptive density estimation using the blockwise Stein method
Philippe Rigollet
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Bernoulli 12(2): 351-370 (April 2006). DOI: 10.3150/bj/1145993978


We study the problem of nonparametric estimation of a probability density of unknown smoothness in L2(R). Expressing mean integrated squared error (MISE) in the Fourier domain, we show that it is close to mean squared error in the Gaussian sequence model. Then applying a modified version of Stein's blockwise method, we obtain a linear monotone oracle inequality. Two consequences of this oracle inequality are that the proposed estimator is sharp minimax adaptive over a scale of Sobolev classes of densities, and that its MISE is asymptotically smaller than or equal to that of kernel density estimators with any bandwidth provided that the kernel belongs to a large class of functions including many standard kernels.


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Philippe Rigollet. "Adaptive density estimation using the blockwise Stein method." Bernoulli 12 (2) 351 - 370, April 2006.


Published: April 2006
First available in Project Euclid: 25 April 2006

zbMATH: 1098.62040
MathSciNet: MR2218559
Digital Object Identifier: 10.3150/bj/1145993978

Keywords: Adaptive density estimation , blockwise Stein rule , kernel oracle , monotone oracle , Oracle inequalities

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


Vol.12 • No. 2 • April 2006
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