Quasi-Concave Density Estimation


Quasi-Concave Density Estimation
Roger Koenker and Ivan Mizera

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.



The paper is available in pdf.

A vignette describing some computational experience with these methods is also available here. This paper is also available from the R package REBayes in .Rnw form so all the code to recreate the figures is visible.

A new review paper is available is now available here. This paper is now forthcoming in Statistical Science. An R package for the methods described in this review can be downloaded in standard compressed R form: MeddeR_0.50.tar.gz. Unfortunately, this package cannot be made available on CRAN due to an incompatibility between ACM-TOMS and CRAN licensing requirements.

Comments are, of course, always welcome.