On the risk of histograms for estimating decreasing densities (Q1103991)

Suppose we want to estimate an element f of the space \(\Theta\) of all decreasing densities on the interval \([\alpha;\alpha +L]\) satisfying \(f(\alpha^+)\leq H\) from n independent observations. We prove that a suitable histogram \(\hat f_ n\) with unequal bin widths will achieve the following risk: \[ \sup_{f\in \Theta}{\mathbb{E}}_ f[\int | \hat f_ n(x)-f(x)| dx]\leq 1.89(S/n)^{1/3}+0.20(S/n)^{2/3}, \] with \(S=Log(HL+1)\). If \(n\geq 39S\), this is only ten times the lower bound given in the author's paper, ibid. 15, 995-1012 (1987; Zbl 0631.62037). An adaptive procedure is suggested when \(\alpha\), L, H are unknown. It is almost as good as the original one.