• Bernoulli
  • Volume 21, Number 4 (2015), 1984-2023.

Pointwise adaptive estimation of a multivariate density under independence hypothesis

Gilles Rebelles

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In this paper, we study the problem of pointwise estimation of a multivariate density. We provide a data-driven selection rule from the family of kernel estimators and derive for it a pointwise oracle inequality. Using the latter bound, we show that the proposed estimator is minimax and minimax adaptive over the scale of anisotropic Nikolskii classes. It is important to emphasize that our estimation method adjusts automatically to eventual independence structure of the underlying density. This, in its turn, allows to reduce significantly the influence of the dimension on the accuracy of estimation (curse of dimensionality). The main technical tools used in our considerations are pointwise uniform bounds of empirical processes developed recently in Lepski [ Math. Methods Statist. 22 (2013) 83–99].

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Bernoulli, Volume 21, Number 4 (2015), 1984-2023.

Received: June 2013
Revised: March 2014
First available in Project Euclid: 5 August 2015

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adaptation density estimation independence structure oracle inequality upper function


Rebelles, Gilles. Pointwise adaptive estimation of a multivariate density under independence hypothesis. Bernoulli 21 (2015), no. 4, 1984--2023. doi:10.3150/14-BEJ633.

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