On the Risk of Histograms for Estimating Decreasing Densities

Suppose we want to estimate an element f of the space Θ of all decreasing densities on the interval a; a + L satisfying f(a⁺) ≤ H from n independent observations. We prove that a suitable histogram f̂ₙ with unequal bin widths will achieve the following risk: f ∈ Θ Ef ∫|f̂ₙ(x) - f(x)|dx ≤ 1.89(S/n)1/3 + 0.20(S/n)2/3, with S = Log(HL + 1). If n ≥ 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.