## Bernoulli

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

### Pointwise adaptive estimation of a multivariate density under independence hypothesis

Gilles Rebelles

#### Abstract

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].

#### Article information

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

Dates
Revised: March 2014
First available in Project Euclid: 5 August 2015

https://projecteuclid.org/euclid.bj/1438777584

Digital Object Identifier
doi:10.3150/14-BEJ633

Mathematical Reviews number (MathSciNet)
MR3378457

Zentralblatt MATH identifier
06502614

#### Citation

Rebelles, Gilles. Pointwise adaptive estimation of a multivariate density under independence hypothesis. Bernoulli 21 (2015), no. 4, 1984--2023. doi:10.3150/14-BEJ633. https://projecteuclid.org/euclid.bj/1438777584

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