On the Risk of Histograms for Estimating Decreasing Densities

Abstract

Suppose we want to estimate an element $f$ of the space $\Theta$ of all decreasing densities on the interval $\lbrack a; a + L \rbrack$ satisfying $f(a^+) \leq H$ from $n$ independent observations. We prove that a suitable histogram $\hat{f}_n$ with unequal bin widths will achieve the following risk: $\sup_{f \in \Theta} \mathbb{E}_f \big\lbrack \int|\hat{f}_n(x) - f(x)|dx \big\rbrack \leq 1.89(S/n)^{1/3} + 0.20(S/n)^{2/3}$, with $S = \operatorname{Log}(HL + 1)$. If $n \geq 39S$, this is only ten times the lower bound given in Birge (1987). An adaptive procedure is suggested when $a, L, H$ are unknown. It is almost as good as the original one.