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September, 1987 Estimating a Density under Order Restrictions: Nonasymptotic Minimax Risk
Lucien Birge
Ann. Statist. 15(3): 995-1012 (September, 1987). DOI: 10.1214/aos/1176350488

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$^{-1}$.

Citation

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Lucien Birge. "Estimating a Density under Order Restrictions: Nonasymptotic Minimax Risk." Ann. Statist. 15 (3) 995 - 1012, September, 1987. https://doi.org/10.1214/aos/1176350488

Information

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

zbMATH: 0631.62037
MathSciNet: MR902241
Digital Object Identifier: 10.1214/aos/1176350488

Subjects:
Primary: 62G05
Secondary: 41A46

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 3 • September, 1987
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