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QUASI-CONCAVE DENSITY ESTIMATION
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DOI：10.1214/10-AOS814.png)
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Abstract
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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.
Keywords:
Density estimation
unimodal
strongly unimodal
shape constraints
convex optimization
duality
entropy
semidefinite programming
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