The Annals of Statistics

Quasi-concave density estimation

Roger Koenker and Ivan Mizera

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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.

Article information

Ann. Statist., Volume 38, Number 5 (2010), 2998-3027.

First available in Project Euclid: 20 August 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62H12: Estimation
Secondary: 62G05: Estimation 62B10: Information-theoretic topics [See also 94A17] 90C25: Convex programming 94A17: Measures of information, entropy

Density estimation unimodal strongly unimodal shape constraints convex optimization duality entropy semidefinite programming


Koenker, Roger; Mizera, Ivan. Quasi-concave density estimation. Ann. Statist. 38 (2010), no. 5, 2998--3027. doi:10.1214/10-AOS814.

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