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October 2010 Quasi-concave density estimation
Roger Koenker, Ivan Mizera
Ann. Statist. 38(5): 2998-3027 (October 2010). DOI: 10.1214/10-AOS814


Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.


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Roger Koenker. Ivan Mizera. "Quasi-concave density estimation." Ann. Statist. 38 (5) 2998 - 3027, October 2010.


Published: October 2010
First available in Project Euclid: 20 August 2010

zbMATH: 1200.62031
MathSciNet: MR2722462
Digital Object Identifier: 10.1214/10-AOS814

Primary: 62G07 , 62H12
Secondary: 62B10 , 62G05 , 90C25 , 94A17

Keywords: Convex optimization , Density estimation , Duality , Entropy , semidefinite programming , shape constraints , strongly unimodal , unimodal

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 5 • October 2010
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