Quasi-concave density estimation
The Annals of Statistics · 2010 · Volume 38 · Issue 5
Abstract
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
62B10, 62G05, 62G07, 62H12, 90C25, 94A17, Convex optimization, Density estimation, Duality, Entropy, semidefinite programming, shape constraints, strongly unimodal, unimodal
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