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December, 1989 The Grenader Estimator: A Nonasymptotic Approach
Lucien Birge
Ann. Statist. 17(4): 1532-1549 (December, 1989). DOI: 10.1214/aos/1176347380


In this paper we shall investigate some nonasymptotic properties of the Grenander estimator of a decreasing density $f$. This estimator is defined as the slope of the smallest concave majorant of the empirical c.d.f. It will be proved that its risk, measured with $\mathbb{L}^1$-loss, is bounded by some functional depending on $f$ and the number $n$ of observations. For classes of uniformly bounded densities with a common compact support, upper bounds for the functional are shown to agree with older results about the minimax risk over these classes. The asymptotic behavior of the functional as $n$ goes to infinity is also in accordance with the known asymptotic performances of the Grenander estimator.


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Lucien Birge. "The Grenader Estimator: A Nonasymptotic Approach." Ann. Statist. 17 (4) 1532 - 1549, December, 1989.


Published: December, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0703.62042
MathSciNet: MR1026298
Digital Object Identifier: 10.1214/aos/1176347380

Primary: 62G05
Secondary: 60E15

Keywords: decreasing densities , Grenander estimator , local nonasymptotic risk

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 4 • December, 1989
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