2015/08/05 by Gilles Rebelles · 21 citations
Engineering · Mathematics · #Adaptive estimator #Advanced Statistical Methods and Models #Control Systems and Identification #Density estimation #Estimator #Independence (probability theory) #Kernel density estimation #Minimax #Minimax estimator #Multivariate kernel density estimation #Pointwise #Statistical Methods and Inference #math.ST #stat.TH
paper · pdf · doi:10.3150/14-bej633
published in Bernoulli 21(4) (Chapman and Hall London) · Published at http://dx.doi.org/10.3150/14-BEJ633 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
openalex publication_date 2015/08/05 · arxiv created 2015/09/18 · arxiv updated 2015/09/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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].