Minimax bounds for Besov classes in density estimation

Abstract

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 ${\mathcal{B}}_{\mathrm{\pi },\mathrm{\infty }}^{\mathit{\alpha }}(R)$. In particular, we may deal with unbounded densities and shed light on the minimax rates of convergence when $\mathrm{\pi }<p$ and $\mathit{\alpha }\in (1∕\mathrm{\pi }-1∕p,1∕\mathrm{\pi }]$.