Estimating a Density under Order Restrictions: Nonasymptotic Minimax Risk

Abstract

Let us consider the class of all unimodal densities defined on some interval of length $L$ and bounded by $H$; we shall study the minimax risk over this class, when we estimate using $n$ i.i.d. observations, the loss being measured by the $\mathbb{L}^1$ distance between the estimator and the true density. We shall prove that if $S = \operatorname{Log}(HL + 1)$, upper and lower bounds for the risk are of the form $C(S/n)^{1/3}$ and the ratio between those bounds is smaller than 44 when $S/n$ is smaller than 220$^{-1}$.