Uncompactly supported density estimation with $ L^{1} $ risk

Kaikai Cao; Youming Liu

doi:10.3934/cpaa.2020177

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2020, Volume 19, Issue 8: 4007-4022. Doi: 10.3934/cpaa.2020177

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Uncompactly supported density estimation with $ L^{1} $ risk

Kaikai Cao^1,2, and
Youming Liu^1, ,

1.
Department of Applied Mathematics, Beijing University of Technology, Pingle Yuan 100, Beijing, 100124, China
2.
School of Mathematics and Information Science, Weifang University, Weifang, Shandong, 261061, China

^* Corresponding author
^* Corresponding author

Received: June 2019

Revised: January 2020

Published: May 2020

Supported by National Natural Science Foundation of China Grant 11771030, and the Science and Technology Program of Beijing Municipal Commission of Education Grant KM202010005025

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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.

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Mathematics Subject Classification: Primary: 62G07, 42C40, 62G20.

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Figure 1. Graph of $ g_l $

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References

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