Minimax bounds for Besov classes in density estimation

We study the problem of density estimation on $[0,1]$ under $\mathbb{L}^p$ norm. We carry out a new piecewise polynomial estimator and prove that it is simultaneously (near)-minimax over a very wide range of Besov classes $\mathcal{B}_{\pi,\infty}^{\alpha}(R)$. In particular, we may deal with unboun...

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Bibliographic Details
Published inElectronic journal of statistics Vol. 15; no. 1; pp. 3184 - 3216
Main Author Sart, Mathieu
Format Journal Article
LanguageEnglish
Published Shaker Heights, OH : Institute of Mathematical Statistics 01.01.2021
Subjects
Mathematics
Statistics
62G07 density estimation
minimax risk
December
2020. 2010 Mathematics Subject Classification. 62G05
Besov spaces
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Summary:We study the problem of density estimation on $[0,1]$ under $\mathbb{L}^p$ norm. We carry out a new piecewise polynomial estimator and prove that it is simultaneously (near)-minimax over a very wide range of Besov classes $\mathcal{B}_{\pi,\infty}^{\alpha}(R)$. In particular, we may deal with unbounded densities and shed light on the minimax rates of convergence when $\pi < p$ and $\alpha \in (1/\pi-1/p, 1/\pi]$.
ISSN:1935-7524
1935-7524
DOI:10.1214/21-EJS1856