About the optimal estimation of a density with infinite support under Hellinger loss

Abstract

The aim of this paper is to give a complete description of the optimal estimation rates for the Hellinger loss when the square root of the density belongs to a Besov ball ${\mathfrak{B}}_{p,\infty }^{\mathit{\alpha }}(R)$. We make them explicit without further conditions when $p<2$, and under a tail dominance condition when p is larger. We also show that these rates can be improved when the density is assumed to be unimodal.