Quasi-concave density estimation

TL;DR

This paper introduces a convex optimization framework for estimating quasi-concave densities, extending log-concave density estimation to broader classes with weaker shape constraints, and explores related entropy-based estimators.

Contribution

It formulates quasi-concave density estimation as a convex optimization problem and establishes duality with Shannon entropy maximization, expanding the scope of shape-constrained density estimation.

Findings

Convex optimization approach for quasi-concave density estimation

Dual formulation links to maximum Shannon entropy

Extension to Renyi and Hellinger discrepancy estimators

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.