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September, 1987 On the Risk of Histograms for Estimating Decreasing Densities
Lucien Birge
Ann. Statist. 15(3): 1013-1022 (September, 1987). DOI: 10.1214/aos/1176350489

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.

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Lucien Birge. "On the Risk of Histograms for Estimating Decreasing Densities." Ann. Statist. 15 (3) 1013 - 1022, September, 1987. https://doi.org/10.1214/aos/1176350489

Information

Published: September, 1987
First available in Project Euclid: 12 April 2007

zbMATH: 0646.62033
MathSciNet: MR902242
Digital Object Identifier: 10.1214/aos/1176350489

Subjects:
Primary: 62G05
Secondary: 62C20

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 3 • September, 1987
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