Uncompactly supported density estimation with <inline-formula><tex-math id="M1">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula> risk

Abstract

The perfect achievements have been made for $ L^{p}\; (1\leq p<+\infty) $ risk estimation, when a density function has compact support. However, there does not exist $ L^{1} $ risk estimation for uncompactly supported densities in general. Motivated by the work of Juditsky & Lambert-Lacroix (A. Juditsky and S. Lambert-Lacroix, On minimax density estimation on $ \mathbb{R} $, Bernoulli, 10(2004), 187-220) and Goldenshluger & Lepski (A. Goldenshluger and O. Lepski, On adaptive minimax density estimation on $ \mathbb{R}^{d} $, Probab. Theory Relat. Fields., 159(2014), 479-543), we provide an adaptive estimate for a family of density functions not necessarily having compact supports in this paper.