Multivariate density estimation under sup-norm loss: Oracle approach, adaptation and independence structure
The Annals of Statistics · 2013 · Volume 41 · Issue 2
Abstract
This paper deals with the density estimation on $\mathbb{R}^{d}$ under sup-norm loss. We provide a fully data-driven estimation procedure and establish for it a so-called sup-norm oracle inequality. The proposed estimator allows us to take into account not only approximation properties of the underlying density, but eventual independence structure as well. Our results contain, as a particular case, the complete solution of the bandwidth selection problem in the multivariate density model. Usefulness of the developed approach is illustrated by application to adaptive estimation over anisotropic Nikolskii classes.
Keywords
62G07, Adaptation, Density estimation, Oracle inequality, upper function
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