  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.
