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{1282315406,
author = {Koenker, Roger and Mizera, Ivan},
title = {Quasi-concave density estimation},
journal = {Ann. Statist.},
volume = {38},
number = {1},
year = {2010},
pages = { 2998-3027},
language = {en},
url = {http://dml.mathdoc.fr/item/1282315406}
}
Koenker, Roger; Mizera, Ivan. Quasi-concave density estimation. Ann. Statist., Tome 38 (2010) no. 1, pp. 2998-3027. http://gdmltest.u-ga.fr/item/1282315406/